The England midfielder is out of contract at Chelsea, who have not yet announced whether their record scorer will be offered a new deal.
In the meantime Sky sources say New York City, part-owned by Manchester City, have made an approach for the 36-year-old, who flew out to Miami with England's World Cup squad on Sunday.
The newly-formed franchise, whose first MLS game is still nine months away, have reportedly also approached Spain striker Villa.
Villa has confirmed he is leaving Atletico Madrid to join an "irresistible project", although he did not name New York as his next club.
Lampard admitted earlier this week that he was unsure about where he would be playing next season, although he made it clear he intended to carry on.
"I don't know if I'll be at Chelsea next season," the former West Ham favourite, 35, told The Sun.
"I'd like to carry on for two or three more years but it would be good to end on a high.
So, if that means it all ends tomorrow, I wouldn't have any regrets.
"You know your shelf-life, so I started to mentally prepare when I hit 30. I've been doing this for a long time, so there's a part of me that's looking forward to the future."