diff --git a/Tests/test_file_tar.py b/Tests/test_file_tar.py
index 78ca0c2787a..39a424c938f 100644
--- a/Tests/test_file_tar.py
+++ b/Tests/test_file_tar.py
@@ -65,3 +65,17 @@ def test_contextmanager() -> None:
     with warnings.catch_warnings(action="error"):
         with TarIO.TarIO(TEST_TAR_FILE, "hopper.jpg"):
             pass
+
+
+@pytest.mark.parametrize("size", (b"-1", b"0"))
+def test_odd(tmp_path: Path, size: bytes) -> None:
+    with open(TEST_TAR_FILE, "rb") as f:
+        data = bytearray(f.read())
+
+    data[124:135] = size.rjust(11)
+
+    tmpfile = tmp_path / "temp.tar"
+    tmpfile.write_bytes(data)
+
+    with pytest.raises(ValueError, match="offset must be positive"):
+        TarIO.TarIO(str(tmpfile), "test")
diff --git a/src/PIL/TarIO.py b/src/PIL/TarIO.py
index 68daf6616a0..487467da422 100644
--- a/src/PIL/TarIO.py
+++ b/src/PIL/TarIO.py
@@ -57,7 +57,11 @@ def __init__(self, tarfile: str, file: str) -> None:
             if file == name:
                 break
 
-            self.fh.seek((size + 511) & (~511), io.SEEK_CUR)
+            offset = (size + 511) & ~511
+            if offset <= 0:
+                msg = "offset must be positive"
+                raise ValueError(msg)
+            self.fh.seek(offset, io.SEEK_CUR)
 
         # Open region
         super().__init__(self.fh, self.fh.tell(), size)