The Ecuadorian's contract expires in six months and the Red Devils are reluctant to extend it.
United have an option to extend Valencia's contract for one more year, and have till March to trigger the extension.
The Red Devils have already used the option to keep David De Gea and Anthony Martial for further 12 months.
But the club have not shown any intention to make an offer to either Valencia or Phil Jones.
And Mirror reports that the 33-year-old defender is considering his options with his representatives already "in negotiations" with Inter, who are interested in landing the versatile player for free in the summer.
Valencia joined United from Wigan in 2009, and has featured in 338 games for the Red Devils during his 10-year career at Old Trafford.
Initially playing as a winger, he has been converted into a first-choice right-back by Sir Alex Ferguson, and given the captain armband by Jose Mourinho at the beginning of the current season.
But following a fallout with the Portuguese manager, and with the emergence of Diogo Dalot, he lost his place in the squad.
After Mourinho's sacking, United caretaker Ole Gunnar Solskjaer started Valencia against Newcastle in his first Premier League appearance since September.
Meanwhile, United have been linked with a shock move for Barcelona flop Philippe Coutinho.
Following his £145million switch from Liverpool, the Brazil international has failed to live up to expectations and the Red Devils are reportedly interested in offering him an escape route.
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