Elmander spent an unsuccessful spell at Bolton, scoring just 18 times in 82 appearances for them after signing from Toulouse for £8.2m in 2008.
But the 32-year-old will now return to the Premier League to bolster manager Chris Hughton's squad, subject to paperwork and international clearance.
"I'm very happy - I'm really looking forward to starting training with the lads and to playing some games also," Elmander told Norwich's official website.
"I know Norwich City has great fans and there's a great atmosphere in the stadium, plus I know a couple of players here already too.
"I'm very glad that the deal has been done and now I'm looking forward to the season ahead."
Elmander has scored 16 goal in 52 games for Galatasaray since joining them on a free transfer from Bolton.
But the Turkish side have deemed him to be surplus to requirements, a decision that suits Hughton.
"I'm really delighted to bring in a player of Johan's experience, not only at club and Premier League level but as a current international," said Hughton.
"It's great to welcome him to our squad."