The 32-year-old former Bolton striker, who signed for the Turkish Super Lig champions in 2011, took to Twitter on Tuesday to confirm reports linking him to Carrow Road.
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Gala chief Bulent Tulun also revealed that the deal is almost done, with agreements having been reached with both the player and the English club.
He told Swedish newspaper Expressen: "I can confirm Johan is moving to England.
"We have agreed with Johan, his new club Norwich and his agent Anders [Wallsten]. We wish Johan the best of luck in the future."
Elmander is set to become Chris Hughton's third addition to his forward line this summer, with Ricky van Wolfswinkel, Gary Hooper and Luciano Becchio already in place following the summer sale of captain Grant Holt to Wigan.