The France international has been in excellent form for the Premier League leaders this season, and Sir Alex Ferguson is keen to reward the fullback with a contract extension.
The 31-year-old has been linked with a move away from Old Trafford, with big-spending Paris St-Germain thought to be interested.
But Ferguson is keen for the left-back to put pen to paper on a new deal. Evra has been first-choice left-back this season, despite the arrival of Alexander Buttner in the summer.
His other competition for the left-back spot, Fabio da Silva, is currently on loan at Queens Park Rangers.
Evra signed for the Red Devils from Monaco in 2006, and his current Manchester United contract is due to expire in May 2014. United hope to extend his deal until the end of the 2014-15 season.
The Frenchman has been in fine goalscoring form for Ferguson's side this term - he's bagged four goals and provided six assists in a season where he's only missed one Premier League game.
United have opened up a 12-point gap at the top of the Premier League table, and will be confident of wrestling back the league crown from last year's winners Manchester City.
But they face a crunch Champions League tie this week against Real Madrid. Evra is in line to start for United in the second-leg last-16 match, as they look to progress after a 1-1 draw at the Bernabeu two weeks ago.