The Telegraph says United have a Friday deadline to use their extension clause.
Valencia, 33, has struggled with injury this season and looks set to leave Old Trafford.
The Ecuadorian began his career at the club as a winger but was moved to full-back.
The Old Trafford skipper's deal runs out in June and has not yet been prolonged by the club.
He has played just once under Ole Gunnar Solskjaer and the time may be right for both parties to move on.