The 20-year-old spent last season on loan with the Canaries' Championship rivals QPR, impressing in his 31 appearances for the Loftus Road-based club.
"I saw him when he was a young kid at Chelsea coming through the reserve team and each time I saw him he seemed to have developed as a player," Grant told Norwich's official website.
"He likes to push up into the opposition box and it gives us a different mix.
"He got six goals for QPR last season and, for someone to come from reserve team football into the Championship, he's done very well.
"We're very pleased to have got him because there were two other Championship clubs that wanted him and we're pleased he's chosen us."