diff --git a/lib/http.c b/lib/http.c
index e748aa55b0..c5accd0f93 100644
--- a/lib/http.c
+++ b/lib/http.c
@@ -3439,6 +3439,7 @@ static CURLcode http_on_response(struct Curl_easy *data,
         /* Switching to HTTP/2, where we will get more responses */
         infof(data, "Received 101, Switching to HTTP/2");
         k->upgr101 = UPGR101_RECEIVED;
+        data->conn->bits.asks_multiplex = FALSE;
         /* We expect more response from HTTP/2 later */
         k->header = TRUE;
         k->headerline = 0; /* restart the header line counter */
@@ -3485,6 +3486,7 @@ static CURLcode http_on_response(struct Curl_easy *data,
   if(k->upgr101 == UPGR101_H2) {
     /* A requested upgrade was denied, poke the multi handle to possibly
        allow a pending pipewait to continue */
+    data->conn->bits.asks_multiplex = FALSE;
     Curl_multi_connchanged(data->multi);
   }
 
diff --git a/lib/http2.c b/lib/http2.c
index fdc8136acf..90ee636230 100644
--- a/lib/http2.c
+++ b/lib/http2.c
@@ -1736,6 +1736,7 @@ CURLcode Curl_http2_request_upgrade(struct dynbuf *req,
   free(base64);
 
   k->upgr101 = UPGR101_H2;
+  data->conn->bits.asks_multiplex = TRUE;
 
   return result;
 }
diff --git a/lib/url.c b/lib/url.c
index 88d3ecc9d3..4794bac8f8 100644
--- a/lib/url.c
+++ b/lib/url.c
@@ -907,7 +907,10 @@ static bool url_match_conn(struct connectdata *conn, void *userdata)
      * connections that do not use this feature */
     return FALSE;
 
-  if(!Curl_conn_is_connected(conn, FIRSTSOCKET)) {
+  if(!Curl_conn_is_connected(conn, FIRSTSOCKET) ||
+     conn->bits.asks_multiplex) {
+    /* Not yet connected, or not yet decided if it multiplexes. The later
+     * happens for HTTP/2 Upgrade: requests that need a response. */
     if(match->may_multiplex) {
       match->seen_pending_conn = TRUE;
       /* Do not pick a connection that has not connected yet */
diff --git a/lib/urldata.h b/lib/urldata.h
index 2a282beb66..48f8a3f70c 100644
--- a/lib/urldata.h
+++ b/lib/urldata.h
@@ -535,6 +535,7 @@ struct ConnectBits {
 #endif
   BIT(bound); /* set true if bind() has already been done on this socket/
                  connection */
+  BIT(asks_multiplex); /* connection asks for multiplexing, but is not yet */
   BIT(multiplex); /* connection is multiplexed */
   BIT(tcp_fastopen); /* use TCP Fast Open */
   BIT(tls_enable_alpn); /* TLS ALPN extension? */