The Sun says Lampard has been told he is no longer wanted at Stamford Bridge, with his contract set to run out in the summer.
United manager Alex Ferguson is convinced the veteran midfielder, 34, can still do a job at the very highest level.
He is a long-time admirer of the player and tried to get him before he even joined Chelsea.
It is unlikely a deal will be done this month — when a fee would be involved — while Lampard is also keen to see out the rest of the season at Stamford Bridge.
But United are prepared to offer him a one-year deal in the summer, with the promise of a second year if everything goes well.
Contact has already been made from Old Trafford to Lampard’s representatives.