England international Ramsdale, 25, made way for the Gunners' new shot-stopper David Raya at the weekend.
Aaron Ramsdale was dropped by Arsenal boss Mikel Arteta on Sunday Credit: Alamy
And the Spaniard, 28, kept a clean sheet as Arsenal claimed a 1-0 victory over Everton at Goodison Park.
The decision to bring in Raya has increased speculation when it comes to who Arteta's No1 will be this season.
According to The Mail, Chelsea could look to take advantage of the situation.
Mauricio Pochettino's first-choice keeper is former Brighton man Robert Sanchez.
But the opportunity to snatch Ramsdale from their London rivals could prove too good to turn down.
A January move is thought to be unlikely, but bids in the region of £60million could convince Arsenal to sell in the summer.
However, Chelsea are not the only club interested in the ex-Sheffield United man.
Bayern Munich are also understood to be monitoring Ramsdale's situation - with his contract running until 2026.
Thomas Tuchel is searching for a long-term replacement to Manuel Neuer, 37.
And Ramsdale may be interested in joining the German champions - who signed fellow England star Harry Kane in the summer.
Ramsdale signed for Arsenal from Sheffield United for £30m two years ago.
He has kept 30 clean sheets in 83 games for Arteta's men, conceding 92 goals.
The Chesterton-born player has also earned four caps for England - the last of which came in a 3-1 win over Scotland earlier this month.
Speaking on Sunday about the situation between Ramsdale and Raya, Arteta said: "No one is guaranteed a spot. I'm not guaranteed a spot.
"We're here to perform at the highest level and to contribute our best capacity to win games. It's no different for anyone.
"If we set ourselves those standards, no matter what happens externally it is not going to be bigger than that. That's the aim we have.
"I haven’t had a single question on why Gabriel Jesus hasn’t started.
"He (Jesus) has won more trophies than anybody else, including me, in that dressing room.
"But I cannot have two players in this position and don’t play them. David has tremendous qualities, like Aaron has, like Karl (Hein) has, but we have to use them. It is like this.
"I am a really young manager. I only have been three and a half years in the job and I have few regrets.
"One of them was that on two occasions I felt after 60 minutes and after 85 minutes, in two games in this period, to change the keeper at that moment and I didn’t do it.
"I didn’t have the courage to do it and I was so unhappy.
"Tell me why not do it. Why not. We have all the qualities in another goalkeeper to do something, something is happening and you want to change momentum.
"Do it. It is a regret that I had. And now my feeling is to get everyone engaged and in the team, they have to play regardless of the competition. That is my message."
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