diff --git a/block/file-posix.c b/block/file-posix.c
index 8cc39a1ef6..9a00d4190a 100644
--- a/block/file-posix.c
+++ b/block/file-posix.c
@@ -1698,7 +1698,11 @@ static int handle_aiocb_write_zeroes_unmap(void *opaque)
 #ifdef CONFIG_FALLOCATE_PUNCH_HOLE
     int ret = do_fallocate(s->fd, FALLOC_FL_PUNCH_HOLE | FALLOC_FL_KEEP_SIZE,
                            aiocb->aio_offset, aiocb->aio_nbytes);
-    if (ret != -ENOTSUP) {
+    switch (ret) {
+    case -ENOTSUP:
+    case -EINVAL:
+        break;
+    default:
         return ret;
     }
 #endif
diff --git a/block/qcow2-cluster.c b/block/qcow2-cluster.c
index 4b5fc8c4a7..a677ba9f5c 100644
--- a/block/qcow2-cluster.c
+++ b/block/qcow2-cluster.c
@@ -1797,8 +1797,15 @@ int qcow2_cluster_zeroize(BlockDriverState *bs, uint64_t offset,
     assert(QEMU_IS_ALIGNED(end_offset, s->cluster_size) ||
            end_offset >= bs->total_sectors << BDRV_SECTOR_BITS);
 
-    /* The zero flag is only supported by version 3 and newer */
+    /*
+     * The zero flag is only supported by version 3 and newer. However, if we
+     * have no backing file, we can resort to discard in version 2.
+     */
     if (s->qcow_version < 3) {
+        if (!bs->backing) {
+            return qcow2_cluster_discard(bs, offset, bytes,
+                                         QCOW2_DISCARD_REQUEST, false);
+        }
         return -ENOTSUP;
     }
 
diff --git a/tests/qemu-iotests/122 b/tests/qemu-iotests/122
index dfd1cd05d6..0f3d4ca851 100755
--- a/tests/qemu-iotests/122
+++ b/tests/qemu-iotests/122
@@ -281,6 +281,36 @@ $QEMU_IMG convert -O $IMGFMT -n "$TEST_IMG" "$TEST_IMG".orig
 
 $QEMU_IMG compare "$TEST_IMG" "$TEST_IMG".orig
 
+echo
+echo '=== -n to an empty image ==='
+echo
+
+TEST_IMG="$TEST_IMG".orig _make_test_img 64M
+
+# Convert with -n, which should not result in a fully allocated image, not even
+# with compat=0.10 (because the target doesn't have a backing file)
+for compat in "1.1" "0.10"; do
+    IMGOPTS="compat=$compat" _make_test_img 64M
+    $QEMU_IMG convert -O $IMGFMT -n "$TEST_IMG".orig "$TEST_IMG"
+    $QEMU_IMG map --output=json "$TEST_IMG"
+done
+
+echo
+echo '=== -n to an empty image with a backing file ==='
+echo
+
+TEST_IMG="$TEST_IMG".orig _make_test_img 64M
+TEST_IMG="$TEST_IMG".base _make_test_img 64M
+
+# Convert with -n, which should still not result in a fully allocated image for
+# compat=1.1 (because it can use zero clusters), but it should be fully
+# allocated with compat=0.10
+for compat in "1.1" "0.10"; do
+    IMGOPTS="compat=$compat" _make_test_img -b "$TEST_IMG".base -F $IMGFMT 64M
+    $QEMU_IMG convert -O $IMGFMT -n "$TEST_IMG".orig "$TEST_IMG"
+    $QEMU_IMG map --output=json "$TEST_IMG"
+done
+
 echo
 echo '=== -n -B to an image without a backing file ==='
 echo
diff --git a/tests/qemu-iotests/122.out b/tests/qemu-iotests/122.out
index f1f195ed77..3a3e121d57 100644
--- a/tests/qemu-iotests/122.out
+++ b/tests/qemu-iotests/122.out
@@ -229,6 +229,23 @@ wrote 65536/65536 bytes at offset 0
 64 KiB, X ops; XX:XX:XX.X (XXX YYY/sec and XXX ops/sec)
 Images are identical.
 
+=== -n to an empty image ===
+
+Formatting 'TEST_DIR/t.IMGFMT.orig', fmt=IMGFMT size=67108864
+Formatting 'TEST_DIR/t.IMGFMT', fmt=IMGFMT size=67108864
+[{ "start": 0, "length": 67108864, "depth": 0, "zero": true, "data": false}]
+Formatting 'TEST_DIR/t.IMGFMT', fmt=IMGFMT size=67108864
+[{ "start": 0, "length": 67108864, "depth": 0, "zero": true, "data": false}]
+
+=== -n to an empty image with a backing file ===
+
+Formatting 'TEST_DIR/t.IMGFMT.orig', fmt=IMGFMT size=67108864
+Formatting 'TEST_DIR/t.IMGFMT.base', fmt=IMGFMT size=67108864
+Formatting 'TEST_DIR/t.IMGFMT', fmt=IMGFMT size=67108864 backing_file=TEST_DIR/t.IMGFMT.base backing_fmt=IMGFMT
+[{ "start": 0, "length": 67108864, "depth": 0, "zero": true, "data": false}]
+Formatting 'TEST_DIR/t.IMGFMT', fmt=IMGFMT size=67108864 backing_file=TEST_DIR/t.IMGFMT.base backing_fmt=IMGFMT
+[{ "start": 0, "length": 67108864, "depth": 0, "zero": false, "data": true, "offset": 327680}]
+
 === -n -B to an image without a backing file ===
 
 Formatting 'TEST_DIR/t.IMGFMT.base', fmt=IMGFMT size=67108864