The club captain is approaching 10 years at Old Trafford but United have not triggered an option on Valencia’s contract that expired on Friday evening.
It is understood the 33-year-old former Wigan player is resigned to leaving the club this summer, but is keen to extend his career.
Valencia is currently at home in Ecuador recovering from a calf injury.
He has made just nine appearances this season and has featured once – at Newcastle on 2 January – since Solskjaer replaced Jose Mourinho in December.
United do have other options at right-back, including Ashley Young, who signed a one-year contract extension last month, and 19-year-old Diogo Dalot, who joined the club from Porto for £19m in June.