The Ivory Coast international was ruled out until the start of next year in pre-season after suffering medial knee ligament damage.
The Sun reports that the Old Trafford hierarchy will open talks with the centre-back over fresh terms, with his current deal set to expire next summer.
The 25-year-old has had bad luck with injuries, having spent four months on the treatment table last season with an ankle problem.
However, Manchester United believe he has a part to play with Ole Gunnar Solskjaer’s side going forward, and they see him as a back-up to Harry Maguire and Victor Lindelof.
Bailly currently earns a reported £120,000-per-week and is understood to have a clause in his contract which allows the Red Devils to extend it by a further two years.
“The club want Eric to stay and are looking for talks about an extension,” an Old Trafford source told the newspaper.
“We don’t want to lose him on a free transfer next summer. It’s just down to both parties agreeing to everything.
“His injury record isn’t a major concern for us. We just think he has been unlucky with everyone over the last few years.
“Ole is a fan of the player and we’re going to convince him to stay – at least that is the plan at present.”
United forked out £26 million to bring the defender in from Villarreal three years ago, and he was linked with several clubs during the summer transfer window.
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