diff --git a/dvc/repo/index.py b/dvc/repo/index.py
index 67692d320..40218c137 100644
--- a/dvc/repo/index.py
+++ b/dvc/repo/index.py
@@ -174,6 +174,30 @@ def _load_data_from_outs(index, prefix, outs):
         )
 
 
+def _load_storage_from_import(storage_map, key, out):
+    from fsspec.utils import tokenize
+
+    from dvc_data.index import FileStorage
+
+    if out.stage.is_db_import:
+        return
+
+    dep = out.stage.deps[0]
+    if not out.hash_info:
+        # partial import
+        fs_cache = out.repo.cache.fs_cache
+        storage_map.add_cache(
+            FileStorage(
+                key,
+                fs_cache.fs,
+                fs_cache.fs.join(fs_cache.path, dep.fs.protocol, tokenize(dep.fs_path)),
+            )
+        )
+
+    if out.stage.is_repo_import or not out.hash_info:
+        storage_map.add_remote(FileStorage(key, dep.fs, dep.fs_path, read_only=True))
+
+
 def _load_storage_from_out(storage_map, key, out):
     from dvc.cachemgr import LEGACY_HASH_NAMES
     from dvc.config import NoRemoteError
@@ -190,36 +214,23 @@ def _load_storage_from_out(storage_map, key, out):
                     path=remote.path,
                     index=remote.index,
                     prefix=(),
+                    read_only=(not out.can_push),
                 )
             )
         else:
             odb = (
                 remote.legacy_odb if out.hash_name in LEGACY_HASH_NAMES else remote.odb
             )
-            storage_map.add_remote(ObjectStorage(key, odb, index=remote.index))
+            storage_map.add_remote(
+                ObjectStorage(
+                    key, odb, index=remote.index, read_only=(not out.can_push)
+                )
+            )
     except NoRemoteError:
         pass
 
-    if out.stage.is_db_import:
-        return
-
     if out.stage.is_import:
-        dep = out.stage.deps[0]
-        if not out.hash_info:
-            from fsspec.utils import tokenize
-
-            # partial import
-            fs_cache = out.repo.cache.fs_cache
-            storage_map.add_cache(
-                FileStorage(
-                    key,
-                    fs_cache.fs,
-                    fs_cache.fs.join(
-                        fs_cache.path, dep.fs.protocol, tokenize(dep.fs_path)
-                    ),
-                )
-            )
-        storage_map.add_remote(FileStorage(key, dep.fs, dep.fs_path, read_only=True))
+        _load_storage_from_import(storage_map, key, out)
 
 
 class Index:
diff --git a/tests/func/test_repo_index.py b/tests/func/test_repo_index.py
index 42a5123a2..22d977d97 100644
--- a/tests/func/test_repo_index.py
+++ b/tests/func/test_repo_index.py
@@ -360,26 +360,47 @@ def test_data_index(tmp_dir, dvc, local_cloud, erepo_dir):
     dvc.imp(os.fspath(erepo_dir), "efoo")
     dvc.imp(os.fspath(erepo_dir), "edir")
 
+    dvc.imp(os.fspath(erepo_dir), "efoo", "efoo_partial", no_download=True)
+    dvc.imp(os.fspath(erepo_dir), "edir", "edir_partial", no_download=True)
+
     local_cloud.gen("ifoo", b"ifoo")
     local_cloud.gen("idir", {"ibar": b"ibar", "isubdir": {"ibaz": b"ibaz"}})
 
     dvc.imp_url(str(local_cloud / "ifoo"))
     dvc.imp_url(str(local_cloud / "idir"))
 
+    dvc.imp_url(str(local_cloud / "ifoo"), "ifoo_partial", no_download=True)
+    dvc.imp_url(str(local_cloud / "idir"), "idir_partial", no_download=True)
+
     index = Index.from_repo(dvc)
     assert index.data_keys == {
         "local": set(),
-        "repo": {("dir",), ("edir",), ("efoo",), ("foo",), ("idir",), ("ifoo",)},
+        "repo": {
+            ("foo",),
+            ("dir",),
+            ("efoo",),
+            ("edir",),
+            ("efoo_partial",),
+            ("edir_partial",),
+            ("ifoo",),
+            ("idir",),
+            ("ifoo_partial",),
+            ("idir_partial",),
+        },
     }
 
     data = index.data["repo"]
     assert set(data.keys()) == {
-        ("dir",),
-        ("edir",),
-        ("efoo",),
         ("foo",),
-        ("idir",),
+        ("dir",),
+        ("efoo",),
+        ("edir",),
+        ("efoo_partial",),
+        ("edir_partial",),
         ("ifoo",),
+        ("idir",),
+        ("ifoo_partial",),
+        ("idir_partial",),
     }
 
     assert not data.storage_map[("foo",)].remote
@@ -387,5 +408,12 @@ def test_data_index(tmp_dir, dvc, local_cloud, erepo_dir):
 
     assert data.storage_map[("efoo",)].remote.read_only
     assert data.storage_map[("edir",)].remote.read_only
-    assert data.storage_map[("ifoo",)].remote.read_only
-    assert data.storage_map[("idir",)].remote.read_only
+
+    assert data.storage_map[("efoo_partial",)].remote.read_only
+    assert data.storage_map[("edir_partial",)].remote.read_only
+
+    assert not data.storage_map[("ifoo",)].remote
+    assert not data.storage_map[("idir",)].remote
+
+    assert data.storage_map[("ifoo_partial",)].remote.read_only
+    assert data.storage_map[("idir_partial",)].remote.read_only