diff --git a/Tests/test_image_convert.py b/Tests/test_image_convert.py
index 1a78f8b4c4c..e573a06a2b2 100644
--- a/Tests/test_image_convert.py
+++ b/Tests/test_image_convert.py
@@ -38,6 +38,12 @@ def convert(im, mode):
             convert(im, output_mode)
 
 
+def test_unsupported_conversion():
+    im = hopper()
+    with pytest.raises(ValueError):
+        im.convert("INVALID")
+
+
 def test_default():
 
     im = hopper("P")
diff --git a/src/PIL/Image.py b/src/PIL/Image.py
index 6ba8e11beaa..ea8aca87800 100644
--- a/src/PIL/Image.py
+++ b/src/PIL/Image.py
@@ -1051,7 +1051,10 @@ def convert_transparency(m, v):
         except ValueError:
             try:
                 # normalize source image and try again
-                im = self.im.convert(getmodebase(self.mode))
+                modebase = getmodebase(self.mode)
+                if modebase == self.mode:
+                    raise
+                im = self.im.convert(modebase)
                 im = im.convert(mode, dither)
             except KeyError as e:
                 raise ValueError("illegal conversion") from e