Keane believes Lampard will be shown the exit door if the Blues’ poor run of form continues.
After spending over £200 million during the summer transfer window and a fairly bright start to the 2020/2021 season, Chelsea have been poor in recent matches.
Lampard’s men have won only one match in their last seven games in all competitions.
Chelsea were comfortably beaten 3-1 by Manchester City in the Premier League at home on Sunday.
Despite the heavy defeat in the hands of Pep Guardiola’s men, Lampard in his post match conference insisted that Chelsea played well in the first ten minutes, a statement which angered Keane.
"Frank said they started the game well, the first 10 minutes. The game is 90 minutes long,” Keane said on Sky Sports.
“I think what will frustrate him most looking back at the game and the run they've been on, for the money they've spent they don't look any stronger than last season.
"It's a new league, new language, even with Covid and no supporters, it's a big ask. But players not sprinting back, I really don't get that. That should be in your instinct as a human being and footballer.
"Frank will be concerned for what he saw today. Today you're thinking with the attacking players, I thought they'd put up a performance, but there was a lack of energy.
"When you're losing a game 3-0 and not putting a tackle in, it's a huge concern. He's got senior players out here, players who have won the World Cup, how they didn't sort it out, making tackles, pulling people in, sitting in. That would be the big concern for me.
"Chelsea don't give managers time. It's in their DNA and history not to give them time."
