Everton fell to a damaging 2-0 defeat at fellow strugglers West Ham on Saturday – a ninth loss in their past 12 Premier League matches.
Everton owner Farhad Moshiri had previously offered public backing for Lampard, 44.
But Moshiri, watching Everton for the first time in more than a year at the West Ham game, was far less supportive when asked after the match whether it was time for Lampard to be sacked.
“I can’t comment. It’s not my decision,” he told Sky Sports.
Speaking about his own future, Lampard said, “Those things are not my choice. It is my job to work, focus and keep my head down.
“I know there’s things going on at the club but it’s never been a consequence for me whether a chairman or board member is at the game.”
Everton felt compelled to act over Lampard’s future as they fight to avoid playing outside the top flight for the first time since 1954.
Lampard arrived at Everton 12 months ago with the team languishing in 16th place and the former Chelsea and England midfielder initially looked a shrewd appointment, helping the Merseyside club avoid relegation.
Fuelled by fervent backing from their fans at Goodison Park, the Toffees beat the drop by winning three of their last six league matches, including a dramatic 3-2 success against Crystal Palace that preserved their top-tier status.
But Everton are now looking for their eighth permanent manager in less than seven years after Lampard failed to build on the momentum from that successful end to last term.
Lampard won only 12 of his 44 matches in charge of Everton in all competitions.
Former Burnley boss Sean Dyche, ex-Leeds coach Marcelo Bielsa and West Ham’s former Everton manager David Moyes are among the potential candidates to replace Lampard, according to bookmakers.
Al-Ittihad boss Nuno Espirito Santo, formerly of Wolves, and ex-Everton striker Wayne Rooney, now in charge of MLS side DC United, have also been mooted as possible contenders.
Hampered by the sale of Brazil forward Richarlison to Tottenham in the close season, Everton managed only three wins in 20 league games this term against a backdrop of mounting fan fury.
There have been widespread protests against the board and Lampard at recent games.
Supporters staged a sit-in demonstration after the recent loss to Southampton, with Everton defender Yerry Mina and team-mate Anthony Gordon confronted by fans when they left Goodison Park in their cars.
Lampard, Chelsea’s all-time leading goalscorer as a player, started his managerial career at Derby.
He led Derby to the Championship play-off final in his only season in charge at Pride Park, which ended in defeat by Aston Villa at Wembley.
Lampard took Chelsea into the Champions League during his one full season as manager at Stamford Bridge, but was dismissed in January 2021 with the Blues in ninth place in the Premier League.