Lampard was sacked by the Blues last week following a poor run of results, however supporters have questioned whether the Chelsea legend should have been given more time.
The Daily Telegraph reports that Lampard departed Stamford Bridge amid a strained relationship with Cech, who is now the technical and performance advisor at the club.
It has also been reported that Lampard had not spoken to club owner Roman Abramovich once before his dismissal after 18 months in charge.
Chelsea's executive director Marina Granovskaia, who helped appoint Lampard in the summer of 2019, was allegedly the person who told the 42-year-old that he had lost his job.
She is also believed to have had a strained relationship with Lampard with the pair disagreeing over club transfers.
The Blues have since appointed former Paris Saint-Germain and Borussia Dortmund boss Thomas Tuchel, who won his first match in charge against Burnley on Sunday.