Inter would like to sign the 33-year-old now but United are reluctant to weaken Ole Gunnar Solskjaer's side.
The Red Devils skipper fell out with sacked Manchester United boss Jose Mourinho earlier in the season.
Inter Milan are reportedly the only party that have assessed his situation and are in touch with his advisers, who know that the United may not end up offering contract extension to the Ecuadorian.
Diogo Dalot signed from Porto last summer and has shown promise in the limited opportunities he has had at United so far, and will likely step into Valencia's shoes at right-back unless United land a big name replacement.
Antonio Valencia joined the club in 2009, under Sir Alex Ferguson and was utilized as a winger on the right flank for most of his 10-year spell at the club.
The Red Devils do have until March to make Valencia an offer, but it's looking unlikely at this current stage.
As for where the captain's armband goes in the event of Valencia's departure, the most likely scenario is that Ashley Young is made club captain but Paul Pogba wears it most weeks, with Young not necessarily an automatic starter.
Valencia has only started in 5 league fixtures this season, but he still remains important for Manchester United who will be left with inexperienced backups at the position if he leaves the club in the next transfer window.