diff --git a/Lib/tarfile.py b/Lib/tarfile.py
index 1d15612616f1d8..043a4ab5a52a6f 100755
--- a/Lib/tarfile.py
+++ b/Lib/tarfile.py
@@ -2345,6 +2345,15 @@ def next(self):
                     raise ReadError(str(e))
             except SubsequentHeaderError as e:
                 raise ReadError(str(e))
+            except Exception as e:
+                try:
+                    import zlib
+                    if isinstance(e, zlib.error):
+                        raise ReadError(f'zlib error: {e}')
+                    else:
+                        raise e
+                except ImportError:
+                    raise e
             break
 
         if tarinfo is not None:
diff --git a/Lib/test/test_tarfile.py b/Lib/test/test_tarfile.py
index 6279309ed3bf1f..06fb97212d8567 100644
--- a/Lib/test/test_tarfile.py
+++ b/Lib/test/test_tarfile.py
@@ -18,6 +18,10 @@
     import gzip
 except ImportError:
     gzip = None
+try:
+    import zlib
+except ImportError:
+    zlib = None
 try:
     import bz2
 except ImportError:
@@ -686,6 +690,16 @@ def test_parallel_iteration(self):
                 self.assertEqual(m1.offset, m2.offset)
                 self.assertEqual(m1.get_info(), m2.get_info())
 
+    @unittest.skipIf(zlib is None, "requires zlib")
+    def test_zlib_error_does_not_leak(self):
+        # bpo-39039: tarfile.open allowed zlib exceptions to bubble up when
+        # parsing certain types of invalid data
+        with unittest.mock.patch("tarfile.TarInfo.fromtarfile") as mock:
+            mock.side_effect = zlib.error
+            with self.assertRaises(tarfile.ReadError):
+                tarfile.open(self.tarname)
+
+
 class MiscReadTest(MiscReadTestBase, unittest.TestCase):
     test_fail_comp = None
 
diff --git a/Misc/NEWS.d/next/Library/2021-08-18-10-36-14.bpo-39039.A63LYh.rst b/Misc/NEWS.d/next/Library/2021-08-18-10-36-14.bpo-39039.A63LYh.rst
new file mode 100644
index 00000000000000..7250055c2a4a9e
--- /dev/null
+++ b/Misc/NEWS.d/next/Library/2021-08-18-10-36-14.bpo-39039.A63LYh.rst
@@ -0,0 +1,2 @@
+tarfile.open raises :exc:`~tarfile.ReadError` when a zlib error occurs
+during file extraction.