diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index 727b5ec92bdc4..f3fc91b331359 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -617,6 +617,7 @@ Indexing
 - Bug in :meth:`DataFrame.reindex` raising ``IndexingError`` wrongly for empty :class:`DataFrame` with ``tolerance`` not None or ``method="nearest"`` (:issue:`27315`)
 - Bug in indexing on a :class:`Series` or :class:`DataFrame` with a :class:`CategoricalIndex` using listlike indexer that contains elements that are in the index's ``categories`` but not in the index itself failing to raise ``KeyError`` (:issue:`37901`)
 - Bug in :meth:`DataFrame.iloc` and :meth:`Series.iloc` aligning objects in ``__setitem__`` (:issue:`22046`)
+- Bug in :meth:`MultiIndex.drop` does not raise if labels are partially found (:issue:`37820`)
 - Bug in :meth:`DataFrame.loc` did not raise ``KeyError`` when missing combination was given with ``slice(None)`` for remaining levels (:issue:`19556`)
 
 Missing
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 9c80236129155..6342e6c793608 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -2084,7 +2084,7 @@ def drop(self, codes, level=None, errors="raise"):
         Parameters
         ----------
         codes : array-like
-            Must be a list of tuples
+            Must be a list of tuples when level is not specified
         level : int or level name, default None
         errors : str, default 'raise'
 
@@ -2139,10 +2139,13 @@ def _drop_from_level(self, codes, level, errors="raise"):
         # are not nan and equal -1, this means they are missing in the index
         nan_codes = isna(codes)
         values[(np.equal(nan_codes, False)) & (values == -1)] = -2
+        if index.shape[0] == self.shape[0]:
+            values[np.equal(nan_codes, True)] = -2
 
+        not_found = codes[values == -2]
+        if len(not_found) != 0 and errors != "ignore":
+            raise KeyError(f"labels {not_found} not found in level")
         mask = ~algos.isin(self.codes[i], values)
-        if mask.all() and errors != "ignore":
-            raise KeyError(f"labels {codes} not found in level")
 
         return self[mask]
 
diff --git a/pandas/tests/indexes/multi/test_drop.py b/pandas/tests/indexes/multi/test_drop.py
index 06019ed0a8b14..c39954b22b0f2 100644
--- a/pandas/tests/indexes/multi/test_drop.py
+++ b/pandas/tests/indexes/multi/test_drop.py
@@ -147,3 +147,24 @@ def test_drop_with_nan_in_index(nulls_fixture):
     msg = r"labels \[Timestamp\('2001-01-01 00:00:00'\)\] not found in level"
     with pytest.raises(KeyError, match=msg):
         mi.drop(pd.Timestamp("2001"), level="date")
+
+
+def test_single_level_drop_partially_missing_elements():
+    # GH 37820
+
+    mi = MultiIndex.from_tuples([(1, 2), (2, 2), (3, 2)])
+    msg = r"labels \[4\] not found in level"
+    with pytest.raises(KeyError, match=msg):
+        mi.drop(4, level=0)
+    with pytest.raises(KeyError, match=msg):
+        mi.drop([1, 4], level=0)
+    msg = r"labels \[nan\] not found in level"
+    with pytest.raises(KeyError, match=msg):
+        mi.drop([np.nan], level=0)
+    with pytest.raises(KeyError, match=msg):
+        mi.drop([np.nan, 1, 2, 3], level=0)
+
+    mi = MultiIndex.from_tuples([(np.nan, 1), (1, 2)])
+    msg = r"labels \['a'\] not found in level"
+    with pytest.raises(KeyError, match=msg):
+        mi.drop([np.nan, 1, "a"], level=0)