The 33-year-old joined the Clarets on a free transfer from Dundee United in June 2008 and made 21 appearances in the Championship last season - including in the play-off final win over Sheffield United - to help the club gain promotion.
However, Kalvenes has started just three matches in the Premier League this campaign, making three further appearances from the bench with his last game in a Burnley shirt coming against Chelsea at the end of January.
But he has nothing but fond memories as he returns to his home city of Bergen in Norway.
Kalvenes told the club's official website: "I've had a fantastic time. If I could pick a time over the last 40 years to be at Burnley I think picked the two best years.
"I would have loved to have played more games this season but looking back at the time since I left Dundee United and coming down here I have experienced more than I could ever have dreamed of. I just have happy memories."