diff --git a/doc/source/whatsnew/v1.0.0.rst b/doc/source/whatsnew/v1.0.0.rst
index a5af4e727391a..3beaa2dfa788a 100644
--- a/doc/source/whatsnew/v1.0.0.rst
+++ b/doc/source/whatsnew/v1.0.0.rst
@@ -147,7 +147,7 @@ Datetimelike
 - Bug in :class:`Timestamp` subtraction when subtracting a :class:`Timestamp` from a ``np.datetime64`` object incorrectly raising ``TypeError`` (:issue:`28286`)
 - Addition and subtraction of integer or integer-dtype arrays with :class:`Timestamp` will now raise ``NullFrequencyError`` instead of ``ValueError`` (:issue:`28268`)
 - Bug in :class:`Series` and :class:`DataFrame` with integer dtype failing to raise ``TypeError`` when adding or subtracting a ``np.datetime64`` object (:issue:`28080`)
--
+- Bug in :class:`Week` with ``weekday`` incorrectly raising ``AttributeError`` instead of ``TypeError`` when adding or subtracting an invalid type (:issue:`28530`)
 
 
 Timedelta
diff --git a/pandas/tests/tseries/offsets/test_offsets.py b/pandas/tests/tseries/offsets/test_offsets.py
index 3ed25b8d3edd5..ddf2c6e65b474 100644
--- a/pandas/tests/tseries/offsets/test_offsets.py
+++ b/pandas/tests/tseries/offsets/test_offsets.py
@@ -4348,3 +4348,12 @@ def test_last_week_of_month_on_offset():
     slow = (ts + offset) - offset == ts
     fast = offset.onOffset(ts)
     assert fast == slow
+
+
+def test_week_add_invalid():
+    # Week with weekday should raise TypeError and _not_ AttributeError
+    #  when adding invalid offset
+    offset = Week(weekday=1)
+    other = Day()
+    with pytest.raises(TypeError, match="Cannot add"):
+        offset + other
diff --git a/pandas/tseries/frequencies.py b/pandas/tseries/frequencies.py
index dfe91b514bbe1..4491e6ad9ac7e 100644
--- a/pandas/tseries/frequencies.py
+++ b/pandas/tseries/frequencies.py
@@ -138,7 +138,7 @@ def to_offset(freq):
                         delta = offset
                     else:
                         delta = delta + offset
-        except Exception:
+        except ValueError:
             raise ValueError(libfreqs.INVALID_FREQ_ERR_MSG.format(freq))
 
     else:
@@ -170,7 +170,7 @@ def to_offset(freq):
                     delta = offset
                 else:
                     delta = delta + offset
-        except Exception:
+        except (ValueError, TypeError):
             raise ValueError(libfreqs.INVALID_FREQ_ERR_MSG.format(freq))
 
     if delta is None:
diff --git a/pandas/tseries/offsets.py b/pandas/tseries/offsets.py
index edf58ba3850a1..82cbfa831bf32 100644
--- a/pandas/tseries/offsets.py
+++ b/pandas/tseries/offsets.py
@@ -605,7 +605,7 @@ def apply(self, other):
             return BDay(self.n, offset=self.offset + other, normalize=self.normalize)
         else:
             raise ApplyTypeError(
-                "Only know how to combine business day with " "datetime or timedelta."
+                "Only know how to combine business day with datetime or timedelta."
             )
 
     @apply_index_wraps
@@ -1545,6 +1545,13 @@ def apply(self, other):
         if self.weekday is None:
             return other + self.n * self._inc
 
+        if not isinstance(other, datetime):
+            raise TypeError(
+                "Cannot add {typ} to {cls}".format(
+                    typ=type(other).__name__, cls=type(self).__name__
+                )
+            )
+
         k = self.n
         otherDay = other.weekday()
         if otherDay != self.weekday: