The France international, whose current deal expires in the summer, is being lined up by the likes of Inter and Monaco but David Moyes is keen for the left-back to stay put
The France international left-back is out of contract in the summer, with United planning to offer a slight increase on his £85,000-a-week deal with the promise of an annual rolling review in an attempt to convince him to stay.
David Moyes's pursuit of Everton defender Leighton Baines last summer appeared to suggest that the Scot did not see a long-term future for the 32-year-old at Old Trafford.
But with Baines deciding to commit his future to the Merseysiders by signing a new deal back in January, the former Everton boss has seemingly had a change of heart and wants Evra to remain at the club.
A United source told Goal: "Moyes is keen to keep Evra beyond the summer but, out of respect for his service to the club, will not stand in his way should he decide that he doesn't want to extend his current deal.
"And despite what has been portrayed in the media, Evra's team-mates remain convinced that he wants to stay and help Moyes through this difficult period in charge of the club."
Goal understands that Inter remain favourites to sign Evra, while Monaco retain an interest in bringing the defender back to France as they continue to build a squad capable of challenging for honours domestically and in Europe.
Should the Frenchman decide to stay at Old Trafford, United's pursuit of Southampton sensation Luke Shaw could have a significant impact on how regularly Moyes uses the 32-year-old next season.
The north west club are expected to be party to a three-way battle to sign the England World Cup hopeful in the summer, with Manchester City and Chelsea also keen on the defender.
Evra, who scored a sensational goal in the Champions League defeat to Bayern Munich, has made 30 league appearances for United this season although his agent's insistence that "things have changed" at the club has done little to reassure supporters that he is planning to stay put.