As it stands, the 34-year-old midfielder will be free to leave west London in the summer, with less than six months to run on his existing deal, and no new offer on the table.
However, Lampard's impressive recent run of form has turned up the pressure on Blues owner Roman Abramovich, to cave in to fans' demands and offer the England international an extension to his £150,000-a-week contract.
"It is my understanding that Frank has been given an indication that the club are to offer him a new deal, the details of which are yet to be discussed," a Chelsea source told the Daily Express.
"Everyone at the club will be delighted if he stays."
Lampard was on the scoresheet again as Chelsea beat Brentford 4-0 in their FA Cup fourth round replay at Stamford Bridge on Sunday, with the talisman netting his 36th goal in the competition, leaving him on 199 goals in total for the Blues - three behind Bobby Tambling's all-time club record.
Supporters have made their thoughts on the contract saga abundantly clear, regularly heard chanting "Sign him up" before, during, and after games. Now, it looks as though they are going to get their wish.
"Frank has so much more to give," said Chelsea team-mate and club captain John Terry in the aftermath of Sunday's game. "He is an inspiration to everyone here.
"He is the best trainer at Chelsea by a million miles. I am not even going to entertain the thought of him playing for another club."
Do YOU want to write for GiveMeSport? Get started today by signing-up and submitting an article HERE: https://www.givemesport.com/writeforgms