The 6ft 7in goalkeeper was at Parkhead on Monday to finalise his move to Glasgow after watching Celtic beat St Mirren 4-0 on Sunday in the Scottish Premier League.
Forster helped Norwich to the League One title on a loan spell last season and Celtic manager Neil Lennon received a glowing recommendation from Canaries boss Paul Lambert, his former midfield partner at Celtic Park.
Lennon has been looking for someone to challenge Lukasz Zaluska since Artur Boruc signed for Fiorentina and the Celtic boss believes he has found him.
Lennon said on Sunday: "He's a big boy, he's come highly recommended, I've watched him myself a couple of times.
"His age, 22, is the only thing going against him because otherwise he's all the qualities you look for in a goalkeeper.
"He's got a massive presence and he's come highly recommended from people in the game whose opinion I trust, so if the medical goes okay, we'll take him on board.
"We've needed competition in that area so he'll add to that."