diff --git a/doc/source/whatsnew/v2.2.0.rst b/doc/source/whatsnew/v2.2.0.rst
index b29d35c8ce332..be565616036e6 100644
--- a/doc/source/whatsnew/v2.2.0.rst
+++ b/doc/source/whatsnew/v2.2.0.rst
@@ -301,8 +301,9 @@ Datetimelike
 ^^^^^^^^^^^^
 - Bug in :meth:`DatetimeIndex.union` returning object dtype for tz-aware indexes with the same timezone but different units (:issue:`55238`)
 - Bug in :meth:`Tick.delta` with very large ticks raising ``OverflowError`` instead of ``OutOfBoundsTimedelta`` (:issue:`55503`)
+- Bug in adding or subtracting a :class:`Week` offset to a ``datetime64`` :class:`Series`, :class:`Index`, or :class:`DataFrame` column with non-nanosecond resolution returning incorrect results (:issue:`55583`)
 - Bug in addition or subtraction of very large :class:`Tick` objects with :class:`Timestamp` or :class:`Timedelta` objects raising ``OverflowError`` instead of ``OutOfBoundsTimedelta`` (:issue:`55503`)
-
+-
 
 Timedelta
 ^^^^^^^^^
diff --git a/pandas/_libs/tslibs/offsets.pyi b/pandas/_libs/tslibs/offsets.pyi
index 1f0fad6abb8b4..aecc8cf681cf8 100644
--- a/pandas/_libs/tslibs/offsets.pyi
+++ b/pandas/_libs/tslibs/offsets.pyi
@@ -277,7 +277,10 @@ def roll_qtrday(
 INVALID_FREQ_ERR_MSG: Literal["Invalid frequency: {0}"]
 
 def shift_months(
-    dtindex: npt.NDArray[np.int64], months: int, day_opt: str | None = ...
+    dtindex: npt.NDArray[np.int64],
+    months: int,
+    day_opt: str | None = ...,
+    reso: int = ...,
 ) -> npt.NDArray[np.int64]: ...
 
 _offset_map: dict[str, BaseOffset]
diff --git a/pandas/_libs/tslibs/offsets.pyx b/pandas/_libs/tslibs/offsets.pyx
index f20073766bc3a..cb2f99eef4226 100644
--- a/pandas/_libs/tslibs/offsets.pyx
+++ b/pandas/_libs/tslibs/offsets.pyx
@@ -3272,7 +3272,8 @@ cdef class Week(SingleConstructorOffset):
     def _apply_array(self, dtarr):
         if self.weekday is None:
             td = timedelta(days=7 * self.n)
-            td64 = np.timedelta64(td, "ns")
+            unit = np.datetime_data(dtarr.dtype)[0]
+            td64 = np.timedelta64(td, unit)
             return dtarr + td64
         else:
             reso = get_unit_from_dtype(dtarr.dtype)
diff --git a/pandas/tests/arithmetic/test_datetime64.py b/pandas/tests/arithmetic/test_datetime64.py
index 67bcafd583086..89df837315396 100644
--- a/pandas/tests/arithmetic/test_datetime64.py
+++ b/pandas/tests/arithmetic/test_datetime64.py
@@ -1415,8 +1415,9 @@ def test_dt64arr_add_sub_relativedelta_offsets(self, box_with_array):
     )
     @pytest.mark.parametrize("normalize", [True, False])
     @pytest.mark.parametrize("n", [0, 5])
+    @pytest.mark.parametrize("unit", ["s", "ms", "us", "ns"])
     def test_dt64arr_add_sub_DateOffsets(
-        self, box_with_array, n, normalize, cls_and_kwargs
+        self, box_with_array, n, normalize, cls_and_kwargs, unit
     ):
         # GH#10699
         # assert vectorized operation matches pointwise operations
@@ -1449,7 +1450,7 @@ def test_dt64arr_add_sub_DateOffsets(
                 Timestamp("2000-05-15"),
                 Timestamp("2001-06-15"),
             ]
-        )
+        ).as_unit(unit)
         vec = tm.box_expected(vec, box_with_array)
         vec_items = vec.iloc[0] if box_with_array is pd.DataFrame else vec
 
@@ -1461,15 +1462,16 @@ def test_dt64arr_add_sub_DateOffsets(
 
         offset = offset_cls(n, normalize=normalize, **kwargs)
 
-        expected = DatetimeIndex([x + offset for x in vec_items])
+        # TODO(GH#55564): as_unit will be unnecessary
+        expected = DatetimeIndex([x + offset for x in vec_items]).as_unit(unit)
         expected = tm.box_expected(expected, box_with_array)
         tm.assert_equal(expected, vec + offset)
 
-        expected = DatetimeIndex([x - offset for x in vec_items])
+        expected = DatetimeIndex([x - offset for x in vec_items]).as_unit(unit)
         expected = tm.box_expected(expected, box_with_array)
         tm.assert_equal(expected, vec - offset)
 
-        expected = DatetimeIndex([offset + x for x in vec_items])
+        expected = DatetimeIndex([offset + x for x in vec_items]).as_unit(unit)
         expected = tm.box_expected(expected, box_with_array)
         tm.assert_equal(expected, offset + vec)
         msg = "(bad|unsupported) operand type for unary"
@@ -2392,7 +2394,8 @@ def test_dti_addsub_object_arraylike(
 
 @pytest.mark.parametrize("years", [-1, 0, 1])
 @pytest.mark.parametrize("months", [-2, 0, 2])
-def test_shift_months(years, months):
+@pytest.mark.parametrize("unit", ["s", "ms", "us", "ns"])
+def test_shift_months(years, months, unit):
     dti = DatetimeIndex(
         [
             Timestamp("2000-01-05 00:15:00"),
@@ -2401,11 +2404,13 @@ def test_shift_months(years, months):
             Timestamp("2000-02-29"),
             Timestamp("2000-12-31"),
         ]
-    )
-    actual = DatetimeIndex(shift_months(dti.asi8, years * 12 + months))
+    ).as_unit(unit)
+    shifted = shift_months(dti.asi8, years * 12 + months, reso=dti._data._creso)
+    shifted_dt64 = shifted.view(f"M8[{dti.unit}]")
+    actual = DatetimeIndex(shifted_dt64)
 
     raw = [x + pd.offsets.DateOffset(years=years, months=months) for x in dti]
-    expected = DatetimeIndex(raw)
+    expected = DatetimeIndex(raw).as_unit(dti.unit)
     tm.assert_index_equal(actual, expected)