The 35-year-old looked set to complete a move to the MLS last summer, before Jose Mourinho arrived and persuaded him to stay at Stamford Bridge.
He has been used sparingly this season however, and LA Galaxy have continued to monitor the situation.
And reports suggest they are ready to offer Lampard a staggering £4million-a-year contract to switch to the MLS.
Chelsea are ready to offer him another one year extension, with the possibility of a coaching role, but Lampard may be ready to accept a new challenge at this stage of his career.
Former Tottenham forward Robbie Keane and former Everton forward Landon Donavon are two of the three ‘designated’ players above the MLS salary cap.
But the club are still on the hunt for a replacement for former England captain David Beckham, and hope Lampard will fill the void at the end of the season.
Meanwhile, reports suggest Chelsea have been offered the chance to sign Bayern Munich forward Mario Mandzukic.
The Croatian international could be on the move in the summer once Robert Lewandowski arrives at the Allianz Arena, and would represent a cheaper alternative to Diego Costa and Radamel Falcao for Jose Mourinho.