The Old Trafford skipper's deal runs out in June and has not yet been prolonged by the club.
Now, the Daily Telegraph is reporting that 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.
He has played just once under Ole Gunnar Solskjaer and the time may be right for both parties to move on.
Portugal youngster Diogo Dalot was signed last summer and is seen as the future in the position.
Full-back has been an issue for United in recent years, with Ashley Young another option for Solskjaer.
But Luke Shaw has finally began to show what he is capable of – making the left flank more secure.
Old Trafford chiefs will spend big this summer and could opt to bring in an experienced full-back ahead of Dalot, 19.
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