Luton have made their ninth summer signing following the arrival of winger Danny Green.
Notts County have signed Togo international Mawouna Amevor on a two-year deal.
The Scottish champions turned down a bid described as 'derisory' on Tuesday, believed to be around £3.5m for the England international.
Forster earned a place in England’s World Cup squad after impressing Roy Hodgson with a string of superb performances in the Champions League for Celtic.
In February this year Forster set a new Scottish league record for consecutive shut-outs, passing former Aberdeen goalkeeper Bobby Clark’s achievement from 1971, by going 13 matches and 1,256 minutes without losing a goal.
The 26-year-old is highly valued by Celtic, and it is expected they would demand at least £10m for his services.
Forster played in both legs of Celtic’s Champions League qualifying tie against KR Reykjavic, keeping two clean sheets as the Glasgow side won 5-0 on aggregate.
In the second leg at Edinburgh's Murrayfield Stadium, the Hoops eased to victory with Virgil van Dijk and Teemu Pukki both scoring twice.
And Pukki is desperate to see the former Newcastle keeper - known affectionately to the Celtic faithful as 'the Great Wall', remain at Celtic Park.
"It's so important for every player to reach the group stages of the Champions League,” Pukki said. "It's the best place to play football so we all want to be there. That's our main aim in the next few weeks.
"And it's important that we keep Fraser because he's one of the best keepers in the world. So we want him to stay. He's one of the most vital parts of this team, he showed that last season and for a few years before.
“He's important to the club. It might be difficult to hold onto him because there are many teams after him. But personally, I hope he stays because he's a great keeper."