The Sun says Cole, 32, is firmly in Ferguson’s sights and both players are out of contract at the end of the season.
Lampard, 34, has already been told that he will not be offered a new deal while Cole has failed to reach an agreement on a new one-year contract at Stamford Bridge.
Ferguson is willing to offer both players a 12-month contract with the promise of a second year if everything goes well.
Lampard is very keen on the move while Cole is thinking about it as the chance to go to big-spending Paris St Germain is also on offer.
That is a wealth of experience leaving the United dressing room and the United boss wants people to replace them who have won the highest honours and can help bring along the younger players.