The Sunday Mirror says the 34-year-old is likely to have to accept a deal heavily based on appearances to stay at Stamford Bridge.
He earns £160,000-a-week, but Chelsea would want to cut his basic salary by as much as half and offer bonuses based on the number of games he plays.
Galaxy were confident of tempting Lampard with a contract worth £8million, until England manager Roy Hodgson urged him to stay in Europe.
Lampard is likely to heed Hodgson's advice and remains desperate to stay at Chelsea.
Sources close to Lampard have previously indicated he would accept a pay cut to finish his career at the Bridge.