The Daily Express says Lampard fears a backlash from Chelsea fans angry at his refusal to sign the £140,000-a-week, four-year deal the club have put on the table.
A source close to the player said: "Frank has had seven wonderful years at the club and does not want to sour relations with Chelsea fans.
"But he fears if he is forced to remain at the club he is going to be given a hard time by some supporters because of the breakdown in talks. What Chelsea fans have to understand, is the club has refused to offer him the five-year deal he wants, and he has always said that if he could not finish his playing career at Chelsea he would like to try his luck abroad.
"Hooking up with Jose Mourinho again is too good an opportunity for him to resist."