The Spaniard has faced criticism for Arsenal's first equalising goal in midweek just days after being at fault for Newcastle's winning goal at the weekend.
The London Evening Standard says Burnley and England keeper Pope has reportedly been flagged as a possible replacement in the summer transfer window should Chelsea decide to cut their losses and make a change.
"I know there's a focus on him at the minute," manager Frank Lampard said of his goalkeeper this week. "That's top-line football. With his feet a couple of times against Arsenal, he took too long and nearly gave it away.
"Sometimes that can change the momentum of a game. That's something obviously that he'll know and has to look at — and we have to look at. It's normal. Any player in a position, particularly goalkeepers, people will look at you and that's when you stand up and show the mettle that you've got."