Lampard will bag up to £4million a year in wages if he moves to LA Galaxy.
Lampard could also bank an extra $1m (£619,000) in merchandise sales, and has been touted as the perfect replacement for Becks. Accepting the deal would make him the highest-paid footballer in MLS.
A source said: "The club feel Frank is what they're missing now David has left.
"He has the profile to fill Beckham's role as the face of the club globally."