Defeat at fellow strugglers West Ham United on Saturday was Everton’s ninth loss in 12 Premier League games.
They are second from bottom of the table with 15 points from 20 matches, above Southampton on goal difference.
Former England midfielder Lampard, 44, replaced Rafael Benitez in January 2022 with the team 16th in the table and helped Everton avoid relegation.
Everton are now looking for their sixth permanent manager in five years.
After a 1-1 draw at reigning champions Manchester City on 31 December, Everton have lost 4-1 to Brighton at Goodison Park, were knocked out of the FA Cup with a 3-1 defeat at Manchester United and were then beaten by Southampton despite taking the lead, before the latest defeat by West Ham.
They have managed only three wins all season.
There have been widespread and vocal protests from fans against the board in recent games, and supporters staged a sit-in demonstration after the loss to Southampton.
Everton’s board of directors missed that game because of what the club claimed was a “real and credible threat to their safety”.
Merseyside Police said no threats or incidents had been reported to officers prior to the fixture before Everton announced “enhanced security procedures” would be put in place.
That was also in response to some supporters confronting defender Yerry Mina and midfielder Anthony Gordon as they left Goodison Park in their cars last weekend.