The England international has even postponed his wedding according to the report as he faces an uncertain summer.
The report claims Lampard, who was due to marry TV presenter Christine Bleakley on June 12, will put off the wedding until his future with Chelsea is sorted.
The 34-year-old is out of contract at Stamford Bridge at the end of the season, and LA Galaxy have increased their offer from $5m a year to $6m as they look to bring in another big name.
The MLS side lost David Beckham last term, and have seen interest and attendances drop as a result. And they're looking to replace the former-Manchester United and Real Madrid man with Lampard.
Lampard himself has been linked with a move to Old Trafford, as it looks increasingly unlikely that Chelsea will offer him a new deal.
But if a move to another Premier League rival doesn't interest the midfielder, LA Galaxy are ready to offer him an escape route to America.
He was an unused substitute for Chelsea's FA Cup semi-final against Manchester City on Sunday, along with John Terry.
The Blues lost 2-1 to City, goals from Sergio Aguero and Samir Nasri enough to book a place in the final against Wigan.