The Frenchman, who was 33 last week, joined United from AS Monaco in 2006 and has made 379 appearances for the club, winning 10 major honours.
He has been first-choice left back since and stood in for Vidic as captain on numerous occasions.
Dutchman Alex Buttner moved to Old Trafford in 2012 as Evra's deputy and British media reports say United have lodged a 27 million pounds ($45.5 million) bid to sign 18-year-old Southampton defender Luke Shaw to fill the position in the long-term.
United are expected to undertake a major overhaul of their squad before next season under the guidance of new manager Louis van Gaal after finishing seventh in the Premier League table a year after winning their 20th English league title.
(Writing by Steve Tongue, additional reporting by Ed Osmond; Editing by Ken Ferris)
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