diff --git a/Tests/test_file_jpeg.py b/Tests/test_file_jpeg.py
index 9f755e8c6..9086c9b0e 100644
--- a/Tests/test_file_jpeg.py
+++ b/Tests/test_file_jpeg.py
@@ -672,8 +672,7 @@ class TestFileJpeg:
             img.save(temp_file, convert_mode=True, fill_color="red")
 
         with Image.open(temp_file) as reloaded:
-            with Image.open("Tests/images/pil123rgba_red.jpg") as target:
-                assert_image_similar(reloaded, target, 4)
+            assert_image_similar_tofile(reloaded, "Tests/images/pil123rgba_red.jpg", 4)
 
     def test_save_tiff_with_dpi(self, tmp_path):
         # Arrange
diff --git a/Tests/test_file_webp.py b/Tests/test_file_webp.py
index 339580bde..e87cc8503 100644
--- a/Tests/test_file_webp.py
+++ b/Tests/test_file_webp.py
@@ -15,8 +15,6 @@ from .helper import (
     skip_unless_feature,
 )
 
-from io import BytesIO
-
 try:
     from PIL import _webp
 
@@ -91,7 +89,7 @@ class TestFileWebp:
             assert_image_similar(image, target, epsilon)
 
     def test_save_convert_mode(self):
-        out = BytesIO()
+        out = io.BytesIO()
         for mode in ["CMYK", "I", "L", "LA", "P"]:
             img = Image.new(mode, (20, 20))
             img.save(out, "WEBP", convert_mode=True)
diff --git a/Tests/test_image.py b/Tests/test_image.py
index 91ecd3c4f..b9d9d7451 100644
--- a/Tests/test_image.py
+++ b/Tests/test_image.py
@@ -7,13 +7,7 @@ import warnings
 
 import pytest
 
-from PIL import (
-    Image,
-    ImageDraw,
-    ImagePalette,
-    TiffImagePlugin,
-    UnidentifiedImageError,
-)
+from PIL import Image, ImageDraw, ImagePalette, TiffImagePlugin, UnidentifiedImageError
 
 from .helper import (
     assert_image_equal,
@@ -138,8 +132,6 @@ class TestImage:
             im.size = (3, 4)
 
     def test_invalid_image(self):
-        import io
-
         im = io.BytesIO(b"")
         with pytest.raises(UnidentifiedImageError):
             with Image.open(im):
@@ -430,14 +422,67 @@ class TestImage:
         for ext in [".cur", ".icns", ".tif", ".tiff"]:
             assert ext in extensions
 
-    def test_no_convert_mode(self, tmp_path):
-        assert not hasattr(TiffImagePlugin, "_convert_mode")
+    def test_supported_modes(self):
+        for format in Image.MIME.keys():
+            try:
+                save_handler = Image.SAVE[format]
+            except KeyError:
+                continue
+            plugin = sys.modules[save_handler.__module__]
+            if not hasattr(plugin, "_supported_modes"):
+                continue
+
+            # Check that the supported modes list is accurate
+            supported_modes = plugin._supported_modes()
+            for mode in [
+                "1",
+                "L",
+                "P",
+                "RGB",
+                "RGBA",
+                "CMYK",
+                "YCbCr",
+                "LAB",
+                "HSV",
+                "I",
+                "F",
+                "LA",
+                "La",
+                "RGBX",
+                "RGBa",
+            ]:
+                out = io.BytesIO()
+                im = Image.new(mode, (100, 100))
+                if mode in supported_modes:
+                    im.save(out, format)
+                else:
+                    with pytest.raises(Exception):
+                        im.save(out, format)
+
+    def test_no_supported_modes_method(self, tmp_path):
+        assert not hasattr(TiffImagePlugin, "_supported_modes")
 
         temp_file = str(tmp_path / "temp.tiff")
 
         im = hopper()
         im.save(temp_file, convert_mode=True)
 
+    def test_convert_mode(self):
+        for mode, modes in [["P", []], ["P", ["P"]]]:  # no modes, same mode
+            im = Image.new(mode, (100, 100))
+            assert im._convert_mode(modes) is None
+
+        for mode, modes in [
+            ["P", ["RGB"]],
+            ["P", ["L"]],  # converting to a non-preferred mode
+            ["LA", ["P"]],
+            ["I", ["L"]],
+            ["RGB", ["L"]],
+            ["RGB", ["CMYK"]],
+        ]:
+            im = Image.new(mode, (100, 100))
+            assert im._convert_mode(modes) is not None
+
     def test_effect_mandelbrot(self):
         # Arrange
         size = (512, 512)
diff --git a/src/PIL/GifImagePlugin.py b/src/PIL/GifImagePlugin.py
index 4de387509..e1d2e82cb 100644
--- a/src/PIL/GifImagePlugin.py
+++ b/src/PIL/GifImagePlugin.py
@@ -1022,11 +1022,8 @@ def getdata(im, offset=(0, 0), **params):
     return fp.data
 
 
-def _convert_mode(im):
-    return {
-        'LA':'P',
-        'CMYK':'RGB'
-    }.get(im.mode)
+def _supported_modes():
+    return ["RGB", "RGBA", "P", "I", "F", "LA", "L", "1"]
 
 
 # --------------------------------------------------------------------
diff --git a/src/PIL/Image.py b/src/PIL/Image.py
index 266044d49..ffe6df0e3 100644
--- a/src/PIL/Image.py
+++ b/src/PIL/Image.py
@@ -2277,16 +2277,18 @@ class Image:
 
         if format.upper() not in SAVE:
             init()
-        if params.pop('save_all', False):
+        if params.pop("save_all", False):
             save_handler = SAVE_ALL[format.upper()]
         else:
             save_handler = SAVE[format.upper()]
 
-        if params.get('convert_mode'):
+        if params.get("convert_mode"):
             plugin = sys.modules[save_handler.__module__]
-            converted_im = self._convert_mode(plugin, params)
-            if converted_im:
-                return converted_im.save(fp, format, **params)
+            if hasattr(plugin, "_supported_modes"):
+                modes = plugin._supported_modes()
+                converted_im = self._convert_mode(modes, params)
+                if converted_im:
+                    return converted_im.save(fp, format, **params)
 
         self.encoderinfo = params
         self.encoderconfig = ()
@@ -2315,32 +2317,57 @@ class Image:
         if open_fp:
             fp.close()
 
-    def _convert_mode(self, plugin, params):
-        if not hasattr(plugin, '_convert_mode'):
+    def _convert_mode(self, modes, params={}):
+        if not modes or self.mode in modes:
             return
-        new_mode = plugin._convert_mode(self)
-        if self.mode == 'LA' and new_mode == 'P':
-            alpha = self.getchannel('A')
+        if self.mode == "P":
+            preferred_modes = []
+            if "A" in self.im.getpalettemode():
+                preferred_modes.append("RGBA")
+            preferred_modes.append("RGB")
+        else:
+            preferred_modes = {
+                "CMYK": ["RGB"],
+                "RGB": ["CMYK"],
+                "RGBX": ["RGB"],
+                "RGBa": ["RGBA", "RGB"],
+                "RGBA": ["RGB"],
+                "LA": ["RGBA", "P", "L"],
+                "La": ["LA", "L"],
+                "L": ["RGB"],
+                "F": ["I"],
+                "I": ["L", "RGB"],
+                "1": ["L"],
+                "YCbCr": ["RGB"],
+                "LAB": ["RGB"],
+                "HSV": ["RGB"],
+            }.get(self.mode, [])
+        for new_mode in preferred_modes:
+            if new_mode in modes:
+                break
+        else:
+            new_mode = modes[0]
+        if self.mode == "LA" and new_mode == "P":
+            alpha = self.getchannel("A")
             # Convert the image into P mode but only use 255 colors
             # in the palette out of 256.
-            im = self.convert('L') \
-                .convert('P', palette=Palette.ADAPTIVE, colors=255)
+            im = self.convert("L").convert("P", palette=Palette.ADAPTIVE, colors=255)
             # Set all pixel values below 128 to 255, and the rest to 0.
             mask = eval(alpha, lambda px: 255 if px < 128 else 0)
             # Paste the color of index 255 and use alpha as a mask.
             im.paste(255, mask)
             # The transparency index is 255.
-            im.info['transparency'] = 255
+            im.info["transparency"] = 255
             return im
 
-        elif self.mode == 'I':
-            im = self.point([i//256 for i in range(65536)], 'L')
-            return im.convert(new_mode) if new_mode != 'L' else im
+        elif self.mode == "I":
+            im = self.point([i // 256 for i in range(65536)], "L")
+            return im.convert(new_mode) if new_mode != "L" else im
 
-        elif self.mode in ('RGBA', 'LA') and new_mode in ('RGB', 'L'):
-            fill_color = params.get('fill_color', 'white')
+        elif self.mode in ("RGBA", "LA") and new_mode in ("RGB", "L"):
+            fill_color = params.get("fill_color", "white")
             background = new(new_mode, self.size, fill_color)
-            background.paste(self, self.getchannel('A'))
+            background.paste(self, self.getchannel("A"))
             return background
 
         elif new_mode:
diff --git a/src/PIL/JpegImagePlugin.py b/src/PIL/JpegImagePlugin.py
index 618b9996a..ebc56ebf3 100644
--- a/src/PIL/JpegImagePlugin.py
+++ b/src/PIL/JpegImagePlugin.py
@@ -819,15 +819,8 @@ def jpeg_factory(fp=None, filename=None):
     return im
 
 
-def _convert_mode(im):
-    mode = im.mode
-    if mode == 'P':
-        return 'RGBA' if 'A' in im.im.getpalettemode() else 'RGB'
-    return {
-        'RGBA':'RGB',
-        'LA':'L',
-        'I':'L'
-    }.get(mode)
+def _supported_modes():
+    return ["RGB", "CMYK", "YCbCr", "RGBX", "L", "1"]
 
 
 # ---------------------------------------------------------------------
diff --git a/src/PIL/PngImagePlugin.py b/src/PIL/PngImagePlugin.py
index 740727bbb..4c3e7143b 100644
--- a/src/PIL/PngImagePlugin.py
+++ b/src/PIL/PngImagePlugin.py
@@ -1420,10 +1420,8 @@ def getchunks(im, **params):
     return fp.data
 
 
-def _convert_mode(im):
-    return {
-        'CMYK':'RGB'
-    }.get(im.mode)
+def _supported_modes():
+    return ["RGB", "RGBA", "P", "I", "LA", "L", "1"]
 
 
 # --------------------------------------------------------------------
diff --git a/src/PIL/WebPImagePlugin.py b/src/PIL/WebPImagePlugin.py
index d81bc99a4..665896f55 100644
--- a/src/PIL/WebPImagePlugin.py
+++ b/src/PIL/WebPImagePlugin.py
@@ -344,17 +344,22 @@ def _save(im, fp, filename):
     fp.write(data)
 
 
-def _convert_mode(im):
-    mode = im.mode
-    if mode == 'P':
-        return 'RGBA' if 'A' in im.im.getpalettemode() else 'RGB'
-    return {
-        # Pillow doesn't support L modes for webp for now.
-        'L':'RGB',
-        'LA':'RGBA',
-        'I':'RGB',
-        'CMYK':'RGB'
-    }.get(mode)
+def _supported_modes():
+    return [
+        "RGB",
+        "RGBA",
+        "RGBa",
+        "RGBX",
+        "CMYK",
+        "YCbCr",
+        "HSV",
+        "I",
+        "F",
+        "P",
+        "LA",
+        "L",
+        "1",
+    ]
 
 
 Image.register_open(WebPImageFile.format, WebPImageFile, _accept)