The 24-year-old failed to make a first-team appearance at Newcastle before his permanent move to Celtic last year, following two loan deals at the Parkhead club.
However, he has forced his way into the England squad this season, thanks in part to the Hoops' remarkable Champions League campaign which saw them make the last-16 before exiting to Juventus in Turin on Wednesday night.
Forster is one of several players who could be subject to bids from interested clubs in the summer.
Speaking ahead of the Clydesdale Bank Premier League match against Ross County in Dingwall tomorrow, he said: "It is definitely one of my ambitions at some point to play down south, I think it is everyone's ambition at some point to test themselves in the Premier League and we will just have to wait and see if it happens.
"We have an ambitious squad and you can't blame them if they want to go and fulfill ambitions elsewhere.
"For me, it is key to stay in and around the England squad.
"I have got the taste of it. I want to push Joe Hart in the long run but Ben Foster has come our of retirement in the last couple of weeks and he is obviously a fantastic keeper.
"But it is important that I am playing and being tested week-in, week-out."