diff --git a/swh/deposit/api/private/deposit_check.py b/swh/deposit/api/private/deposit_check.py
--- a/swh/deposit/api/private/deposit_check.py
+++ b/swh/deposit/api/private/deposit_check.py
@@ -54,7 +54,7 @@
 
 
 class APIChecks(APIPrivateView, APIGet, DepositReadMixin):
-    """Dedicated class to read a deposit's raw archives content.
+    """Dedicated class to trigger the deposit checks on deposit archives and metadata.
 
     Only GET is supported.
 
@@ -67,7 +67,7 @@
             The deposit to check archives for
 
         Returns
-            tuple (status, error_detail): True, None if all archives
+            tuple (status, error_or_warning_details): True, None if all archives
             are ok, (False, <detailed-error>) otherwise.
 
         """
@@ -133,8 +133,14 @@
     def process_get(
         self, req: Request, collection_name: str, deposit: Deposit
     ) -> Tuple[int, Dict, str]:
-        """Build a unique tarball from the multiple received and stream that
-           content to the client.
+        """Trigger the checks on the deposit archives and then on the deposit metadata. If any
+        problems or warnings are raises, the deposit status is updated accordingly
+        (verified or rejected) with details.
+
+        Note that if all checks are ok, the deposit status is updated to the verified
+        status and a loading task is scheduled for the deposit to be ingested.
+        Otherwise, the deposit is marked as rejected with the problems' details. A json
+        response is returned to the caller with the deposit details.
 
         Args:
             req: Client request
@@ -142,45 +148,48 @@
             deposit: Deposit concerned by the reading
 
         Returns:
-            Tuple status, stream of content, content-type
+            Tuple (status, json response, content-type)
 
         """
         metadata, _ = self._metadata_get(deposit)
-        problems: Dict = {}
+        problems_or_warnings: Dict = {}
         # will check each deposit's associated request (both of type
         # archive and metadata) for errors
-        archives_status, error_detail = self._check_deposit_archives(deposit)
-        if not archives_status:
-            assert error_detail is not None
-            problems.update(error_detail)
-
-        metadata_status, error_detail = check_metadata(metadata)
-        if not metadata_status:
-            assert error_detail is not None
-            problems.update(error_detail)
-
-        deposit_status = archives_status and metadata_status
-
-        # if any problems arose, the deposit is rejected
-        if not deposit_status:
-            deposit.status = DEPOSIT_STATUS_REJECTED
-            deposit.status_detail = problems
-            response = {
-                "status": deposit.status,
-                "details": deposit.status_detail,
-            }
-        else:
-            deposit.status = DEPOSIT_STATUS_VERIFIED
-            response = {
-                "status": deposit.status,
-            }
-            if not deposit.load_task_id and self.config["checks"]:
-                url = deposit.origin_url
-                task = create_oneshot_task_dict(
-                    "load-deposit", url=url, deposit_id=deposit.id, retries_left=3
-                )
-                load_task_id = self.scheduler.create_tasks([task])[0]["id"]
-                deposit.load_task_id = load_task_id
+        archives_status_ok, error_or_warning_details = self._check_deposit_archives(
+            deposit
+        )
+        if not archives_status_ok:
+            assert error_or_warning_details is not None
+            problems_or_warnings.update(error_or_warning_details)
+
+        metadata_status_ok, error_or_warning_details = check_metadata(metadata)
+        # Ensure in case of error, we do have the rejection details
+        assert metadata_status_ok or (
+            not metadata_status_ok and error_or_warning_details is not None
+        )
+        # we can have warnings even if checks are ok (e.g. missing suggested field)
+        problems_or_warnings.update(error_or_warning_details or {})
+
+        deposit_status_ok = archives_status_ok and metadata_status_ok
+        # if any problems_or_warnings arose, the deposit is rejected
+        deposit.status = (
+            DEPOSIT_STATUS_VERIFIED if deposit_status_ok else DEPOSIT_STATUS_REJECTED
+        )
+        response: Dict = {
+            "status": deposit.status,
+        }
+        if problems_or_warnings:
+            deposit.status_detail = problems_or_warnings
+            response["details"] = problems_or_warnings
+
+        # Deposit ok, then we schedule the deposit loading task (if not already done)
+        if deposit_status_ok and not deposit.load_task_id and self.config["checks"]:
+            url = deposit.origin_url
+            task = create_oneshot_task_dict(
+                "load-deposit", url=url, deposit_id=deposit.id, retries_left=3
+            )
+            load_task_id = self.scheduler.create_tasks([task])[0]["id"]
+            deposit.load_task_id = load_task_id
 
         deposit.save()
 
diff --git a/swh/deposit/tests/api/test_deposit_private_check.py b/swh/deposit/tests/api/test_deposit_private_check.py
--- a/swh/deposit/tests/api/test_deposit_private_check.py
+++ b/swh/deposit/tests/api/test_deposit_private_check.py
@@ -59,6 +59,14 @@
         deposit = Deposit.objects.get(pk=deposit.id)
         assert deposit.status == DEPOSIT_STATUS_VERIFIED
 
+        # Deposit is ok but it's missing suggested fields in its metadata detected by
+        # the checks
+        status_detail = deposit.status_detail["metadata"]
+        assert len(status_detail) == 1
+        suggested = status_detail[0]
+        assert suggested["summary"] == SUGGESTED_FIELDS_MISSING
+        assert set(suggested["fields"]) == set([METADATA_PROVENANCE_KEY])
+
         deposit.status = DEPOSIT_STATUS_DEPOSITED
         deposit.save()