diff --git a/Lib/test/test_bytes.py b/Lib/test/test_bytes.py
index 5c6238b..234b56c 100644
--- a/Lib/test/test_bytes.py
+++ b/Lib/test/test_bytes.py
@@ -305,6 +305,11 @@
         self.assertTrue(b.startswith(b"h"))
         self.assertFalse(b.startswith(b"hellow"))
         self.assertFalse(b.startswith(b"ha"))
+        with self.assertRaises(TypeError) as cm:
+            b.startswith([b'h'])
+        exc = str(cm.exception)
+        self.assertIn('bytes', exc)
+        self.assertIn('tuple', exc)
 
     def test_endswith(self):
         b = self.type2test(b'hello')
@@ -314,6 +319,11 @@
         self.assertTrue(b.endswith(b"o"))
         self.assertFalse(b.endswith(b"whello"))
         self.assertFalse(b.endswith(b"no"))
+        with self.assertRaises(TypeError) as cm:
+            b.endswith([b'o'])
+        exc = str(cm.exception)
+        self.assertIn('bytes', exc)
+        self.assertIn('tuple', exc)
 
     def test_find(self):
         b = self.type2test(b'mississippi')
diff --git a/Lib/test/test_unicode.py b/Lib/test/test_unicode.py
index ace3736..f35458a 100644
--- a/Lib/test/test_unicode.py
+++ b/Lib/test/test_unicode.py
@@ -819,6 +819,14 @@
         self.assertEqual('%f' % INF, 'inf')
         self.assertEqual('%F' % INF, 'INF')
 
+    def test_startswith_endswith_errors(self):
+        for meth in ('foo'.startswith, 'foo'.endswith):
+            with self.assertRaises(TypeError) as cm:
+                meth(['f'])
+            exc = str(cm.exception)
+            self.assertIn('str', exc)
+            self.assertIn('tuple', exc)
+
     @support.run_with_locale('LC_ALL', 'de_DE', 'fr_FR')
     def test_format_float(self):
         # should not format with a comma, but always with C locale
diff --git a/Misc/NEWS b/Misc/NEWS
index bba2ac6..822569a 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -10,6 +10,9 @@
 Core and Builtins
 -----------------
 
+- Issue #6780: fix starts/endswith error message to mention that tuples are
+  accepted too.
+
 - Issue #5057: fix a bug in the peepholer that led to non-portable pyc files
   between narrow and wide builds while optimizing BINARY_SUBSCR on non-BMP
   chars (e.g. "\U00012345"[0]).
diff --git a/Objects/bytearrayobject.c b/Objects/bytearrayobject.c
index 970a6b5..fba5758 100644
--- a/Objects/bytearrayobject.c
+++ b/Objects/bytearrayobject.c
@@ -1304,7 +1304,7 @@
 Return True if B starts with the specified prefix, False otherwise.\n\
 With optional start, test B beginning at that position.\n\
 With optional end, stop comparing B at that position.\n\
-prefix can also be a tuple of strings to try.");
+prefix can also be a tuple of bytes to try.");
 
 static PyObject *
 bytearray_startswith(PyByteArrayObject *self, PyObject *args)
@@ -1331,8 +1331,12 @@
         Py_RETURN_FALSE;
     }
     result = _bytearray_tailmatch(self, subobj, start, end, -1);
-    if (result == -1)
+    if (result == -1) {
+        if (PyErr_ExceptionMatches(PyExc_TypeError))
+            PyErr_Format(PyExc_TypeError, "startswith first arg must be bytes "
+                         "or a tuple of bytes, not %s", Py_TYPE(subobj)->tp_name);
         return NULL;
+    }
     else
         return PyBool_FromLong(result);
 }
@@ -1343,7 +1347,7 @@
 Return True if B ends with the specified suffix, False otherwise.\n\
 With optional start, test B beginning at that position.\n\
 With optional end, stop comparing B at that position.\n\
-suffix can also be a tuple of strings to try.");
+suffix can also be a tuple of bytes to try.");
 
 static PyObject *
 bytearray_endswith(PyByteArrayObject *self, PyObject *args)
@@ -1370,8 +1374,12 @@
         Py_RETURN_FALSE;
     }
     result = _bytearray_tailmatch(self, subobj, start, end, +1);
-    if (result == -1)
+    if (result == -1) {
+        if (PyErr_ExceptionMatches(PyExc_TypeError))
+            PyErr_Format(PyExc_TypeError, "endswith first arg must be bytes or "
+                         "a tuple of bytes, not %s", Py_TYPE(subobj)->tp_name);
         return NULL;
+    }
     else
         return PyBool_FromLong(result);
 }
diff --git a/Objects/bytesobject.c b/Objects/bytesobject.c
index 7561ae5..ea14be6 100644
--- a/Objects/bytesobject.c
+++ b/Objects/bytesobject.c
@@ -2224,8 +2224,12 @@
         Py_RETURN_FALSE;
     }
     result = _bytes_tailmatch(self, subobj, start, end, -1);
-    if (result == -1)
+    if (result == -1) {
+        if (PyErr_ExceptionMatches(PyExc_TypeError))
+            PyErr_Format(PyExc_TypeError, "startswith first arg must be bytes "
+                         "or a tuple of bytes, not %s", Py_TYPE(subobj)->tp_name);
         return NULL;
+    }
     else
         return PyBool_FromLong(result);
 }
@@ -2264,8 +2268,12 @@
         Py_RETURN_FALSE;
     }
     result = _bytes_tailmatch(self, subobj, start, end, +1);
-    if (result == -1)
+    if (result == -1) {
+        if (PyErr_ExceptionMatches(PyExc_TypeError))
+            PyErr_Format(PyExc_TypeError, "endswith first arg must be bytes or "
+                         "a tuple of bytes, not %s", Py_TYPE(subobj)->tp_name);
         return NULL;
+    }
     else
         return PyBool_FromLong(result);
 }
diff --git a/Objects/unicodeobject.c b/Objects/unicodeobject.c
index efe6879..22d2137 100644
--- a/Objects/unicodeobject.c
+++ b/Objects/unicodeobject.c
@@ -9101,8 +9101,12 @@
         Py_RETURN_FALSE;
     }
     substring = (PyUnicodeObject *)PyUnicode_FromObject(subobj);
-    if (substring == NULL)
+    if (substring == NULL) {
+        if (PyErr_ExceptionMatches(PyExc_TypeError))
+            PyErr_Format(PyExc_TypeError, "startswith first arg must be str or "
+                         "a tuple of str, not %s", Py_TYPE(subobj)->tp_name);
         return NULL;
+    }
     result = tailmatch(self, substring, start, end, -1);
     Py_DECREF(substring);
     return PyBool_FromLong(result);
@@ -9145,9 +9149,12 @@
         Py_RETURN_FALSE;
     }
     substring = (PyUnicodeObject *)PyUnicode_FromObject(subobj);
-    if (substring == NULL)
+    if (substring == NULL) {
+        if (PyErr_ExceptionMatches(PyExc_TypeError))
+            PyErr_Format(PyExc_TypeError, "endswith first arg must be str or "
+                         "a tuple of str, not %s", Py_TYPE(subobj)->tp_name);
         return NULL;
-
+    }
     result = tailmatch(self, substring, start, end, +1);
     Py_DECREF(substring);
     return PyBool_FromLong(result);