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.