The Sun says Lampard is safe in the job for now but Tuesday's abject defeat at Leicester has fuelled growing concerns that the axe is looming.
Upcoming games against out-of-form Wolves and struggling Burnley have now become pivotal matches instead of routine rollovers for the head coach.
Then comes a daunting trip to Tottenham on February 4, which could also have a huge bearing on Lampard's reign.
Despite spending £217.3million on new players, Chelsea have slipped to eighth in the Premier League, losing five of their last eight games.
Lampard, 42, is running out of time to turn things around and facing an increasingly uphill battle to convince owner Roman Abramovich he can lead the team, for whom he scored a record 211 goals as a player.
Club insiders fear the huge, rapid investment in new talent has given Lampard too much choice in his squad and that has led to problems.
There are also claims workaholic Lamps' training can be too intense.