The French full-back's current deal is due to expire at the end of the season.
United are reluctant to part with Evra's considerable experience, with the 32-year-old having spent the last seven years at Old Trafford - during which time he has helped to land five Premier League titles and a UEFA Champions League crown.
David Moyes was, however, heavily linked with a couple of high-profile left-backs over the summer, with Everton's Leighton Baines and Real Madrid's Fabio Coentrao on his transfer radar.
He may be forced to step up that hunt in January, with Evra revealing that he is yet to agree fresh terms.
Speculation has suggested for some time that he could be tempted to return to France before hanging up his boots - with former employers Monaco and big-spending Paris St Germain reported to be keen.
Evra has offered no indication as to what his next move will be, but it appears that his future past the end of 2013/14 lies outside of England.
Asked by Telefoot in France if he is concerned about his expiring contract at United, he said: "No, not at all.
"It is a personal thing. It is not that Manchester United do not want me to sign a new contract.
"Manchester United would like me to finish my career there, but this is something personal."