diff --git a/pandas/tests/tseries/offsets/test_offsets.py b/pandas/tests/tseries/offsets/test_offsets.py
index 08dbc1345b9d4..dfa2a7a67b877 100644
--- a/pandas/tests/tseries/offsets/test_offsets.py
+++ b/pandas/tests/tseries/offsets/test_offsets.py
@@ -51,7 +51,6 @@
     CustomBusinessMonthBegin,
     CustomBusinessMonthEnd,
     DateOffset,
-    Day,
     Easter,
     FY5253Quarter,
     LastWeekOfMonth,
@@ -798,65 +797,6 @@ def test_tick_normalize_raises(tick_classes):
         cls(n=3, normalize=True)
 
 
-def test_weeks_onoffset():
-    # GH#18510 Week with weekday = None, normalize = False should always
-    # be is_on_offset
-    offset = Week(n=2, weekday=None)
-    ts = Timestamp("1862-01-13 09:03:34.873477378+0210", tz="Africa/Lusaka")
-    fast = offset.is_on_offset(ts)
-    slow = (ts + offset) - offset == ts
-    assert fast == slow
-
-    # negative n
-    offset = Week(n=2, weekday=None)
-    ts = Timestamp("1856-10-24 16:18:36.556360110-0717", tz="Pacific/Easter")
-    fast = offset.is_on_offset(ts)
-    slow = (ts + offset) - offset == ts
-    assert fast == slow
-
-
-def test_weekofmonth_onoffset():
-    # GH#18864
-    # Make sure that nanoseconds don't trip up is_on_offset (and with it apply)
-    offset = WeekOfMonth(n=2, week=2, weekday=0)
-    ts = Timestamp("1916-05-15 01:14:49.583410462+0422", tz="Asia/Qyzylorda")
-    fast = offset.is_on_offset(ts)
-    slow = (ts + offset) - offset == ts
-    assert fast == slow
-
-    # negative n
-    offset = WeekOfMonth(n=-3, week=1, weekday=0)
-    ts = Timestamp("1980-12-08 03:38:52.878321185+0500", tz="Asia/Oral")
-    fast = offset.is_on_offset(ts)
-    slow = (ts + offset) - offset == ts
-    assert fast == slow
-
-
-def test_last_week_of_month_on_offset():
-    # GH#19036, GH#18977 _adjust_dst was incorrect for LastWeekOfMonth
-    offset = LastWeekOfMonth(n=4, weekday=6)
-    ts = Timestamp("1917-05-27 20:55:27.084284178+0200", tz="Europe/Warsaw")
-    slow = (ts + offset) - offset == ts
-    fast = offset.is_on_offset(ts)
-    assert fast == slow
-
-    # negative n
-    offset = LastWeekOfMonth(n=-4, weekday=5)
-    ts = Timestamp("2005-08-27 05:01:42.799392561-0500", tz="America/Rainy_River")
-    slow = (ts + offset) - offset == ts
-    fast = offset.is_on_offset(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
-
-
 @pytest.mark.parametrize(
     "attribute",
     [
diff --git a/pandas/tests/tseries/offsets/test_week.py b/pandas/tests/tseries/offsets/test_week.py
index b46a36e00f2da..be574fd963eff 100644
--- a/pandas/tests/tseries/offsets/test_week.py
+++ b/pandas/tests/tseries/offsets/test_week.py
@@ -1,5 +1,8 @@
 """
-Tests for offset.Week, offset.WeekofMonth and offset.LastWeekofMonth
+Tests for the following offsets:
+- Week
+- WeekOfMonth
+- LastWeekOfMonth
 """
 from datetime import (
     datetime,
@@ -10,6 +13,7 @@
 
 from pandas._libs.tslibs import Timestamp
 from pandas._libs.tslibs.offsets import (
+    Day,
     LastWeekOfMonth,
     Week,
     WeekOfMonth,
@@ -121,6 +125,30 @@ def test_is_on_offset(self, weekday):
                 expected = False
         assert_is_on_offset(offset, date, expected)
 
+    @pytest.mark.parametrize(
+        "n,date",
+        [
+            (2, "1862-01-13 09:03:34.873477378+0210"),
+            (-2, "1856-10-24 16:18:36.556360110-0717"),
+        ],
+    )
+    def test_is_on_offset_weekday_none(self, n, date):
+        # GH 18510 Week with weekday = None, normalize = False
+        # should always be is_on_offset
+        offset = Week(n=n, weekday=None)
+        ts = Timestamp(date, tz="Africa/Lusaka")
+        fast = offset.is_on_offset(ts)
+        slow = (ts + offset) - offset == ts
+        assert fast == slow
+
+    def test_week_add_invalid(self):
+        # 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
+
 
 class TestWeekOfMonth(Base):
     _offset = WeekOfMonth
@@ -221,6 +249,22 @@ def test_is_on_offset(self, case):
         offset = WeekOfMonth(week=week, weekday=weekday)
         assert offset.is_on_offset(dt) == expected
 
+    @pytest.mark.parametrize(
+        "n,week,date,tz",
+        [
+            (2, 2, "1916-05-15 01:14:49.583410462+0422", "Asia/Qyzylorda"),
+            (-3, 1, "1980-12-08 03:38:52.878321185+0500", "Asia/Oral"),
+        ],
+    )
+    def test_is_on_offset_nanoseconds(self, n, week, date, tz):
+        # GH 18864
+        # Make sure that nanoseconds don't trip up is_on_offset (and with it apply)
+        offset = WeekOfMonth(n=n, week=week, weekday=0)
+        ts = Timestamp(date, tz=tz)
+        fast = offset.is_on_offset(ts)
+        slow = (ts + offset) - offset == ts
+        assert fast == slow
+
 
 class TestLastWeekOfMonth(Base):
     _offset = LastWeekOfMonth
@@ -298,6 +342,21 @@ def test_is_on_offset(self, case):
         offset = LastWeekOfMonth(weekday=weekday)
         assert offset.is_on_offset(dt) == expected
 
+    @pytest.mark.parametrize(
+        "n,weekday,date,tz",
+        [
+            (4, 6, "1917-05-27 20:55:27.084284178+0200", "Europe/Warsaw"),
+            (-4, 5, "2005-08-27 05:01:42.799392561-0500", "America/Rainy_River"),
+        ],
+    )
+    def test_last_week_of_month_on_offset(self, n, weekday, date, tz):
+        # GH 19036, GH 18977 _adjust_dst was incorrect for LastWeekOfMonth
+        offset = LastWeekOfMonth(n=n, weekday=weekday)
+        ts = Timestamp(date, tz=tz)
+        slow = (ts + offset) - offset == ts
+        fast = offset.is_on_offset(ts)
+        assert fast == slow
+
     def test_repr(self):
         assert (
             repr(LastWeekOfMonth(n=2, weekday=1)) == "<2 * LastWeekOfMonths: weekday=1>"