Lampard was axed on Monday afternoon following a woeful run of results which has seen Chelsea slip down to ninth in the Premier League table.
The Athletic says one of the reasons for Lampard's dismissal was due to his relationship with influential director Granovskaia becoming fractured after several disagreements.
The real bone of contention was when the manager began to drop the club's £72million record-signing Kepa towards the backend of last season after he had made a number of high-profile errors.
At the end of the last campaign, Lampard requested funds to sign a new keeper despite Chelsea chiefs reportedly telling him he should put his arm around Kepa and try to rebuild his confidence.
Kepa was unhappy with how Lampard dealt with the situation and is said to have felt that he did not provide sufficient support - and the Chelsea boss eventually got his way, replacing him as the club's No 1 by signing Edouard Mendy from Rennes.
Matters deteriorated further when Lampard continued to push to sign Declan Rice from West Ham at the annoyance of his bosses. A number of key figures at the club had reservations about the England international while the same report claim the club were 'wary' of the potential embarrassment of buying back a player who they had released as a teenager in a deal worth in excess of £70million.
Lampard was determined to acquire Rice and also Burnley's James Tarkowski and continually tried to persuade the club to broker a deal, but Granovskaia was not keen on either player and in the end neither joined.