diff --git a/docs/configuration.rst b/docs/configuration.rst
index f7a1bbcc..8a0b991b 100644
--- a/docs/configuration.rst
+++ b/docs/configuration.rst
@@ -3002,15 +3002,19 @@ Description
 extractor.twitter.conversations
 -------------------------------
 Type
-    ``bool``
+    * ``bool``
+    * ``string``
 Default
     ``false``
 Description
     For input URLs pointing to a single Tweet,
     e.g. `https://twitter.com/i/web/status/<TweetID>`,
     fetch media from all Tweets and replies in this `conversation
-    <https://help.twitter.com/en/using-twitter/twitter-conversations>`__
-    or thread.
+    <https://help.twitter.com/en/using-twitter/twitter-conversations>`__.
+
+    If this option is equal to ``"accessible"``,
+    only download from conversation Tweets
+    if the given initial Tweet is accessible.
 
 
 extractor.twitter.csrf
diff --git a/gallery_dl/extractor/twitter.py b/gallery_dl/extractor/twitter.py
index 710bde33..10db9744 100644
--- a/gallery_dl/extractor/twitter.py
+++ b/gallery_dl/extractor/twitter.py
@@ -919,7 +919,9 @@ Your reaction.""",
         self.tweet_id = match.group(2)
 
     def tweets(self):
-        if self.config("conversations", False):
+        conversations = self.config("conversations")
+        if conversations:
+            self._accessible = (conversations == "accessible")
             return self._tweets_conversation(self.tweet_id)
         else:
             return self._tweets_single(self.tweet_id)
@@ -950,6 +952,11 @@ Your reaction.""",
                     tweet.get("_retweet_id_str") == tweet_id:
                 self._assign_user(tweet["core"]["user_results"]["result"])
                 break
+        else:
+            # initial Tweet not accessible
+            if self._accessible:
+                return ()
+            return buffer
 
         return itertools.chain(buffer, tweets)