The goalkeeper is on loan at the Scottish giants after falling out of favour at Southampton.
Forster said: "I'd love to be a part of the England squad.
"I'd never close that door. I'd always want that to be an ambition of mine.
"We went through the Champions League the last time, we're going through the Europa League this year.
"It all comes down to me performing the best I can, whether it's Europa League or in the league.
"I don't know Gareth Southgate's preferences. But play as well as I can and the rest will take care of itself.
"If I'm doing that then it forces other people into making decisions. The main thing is I work hard and try to perform well here."