The Slovenia international, Hull's club captain, was a driving force in their return to the Premier League last season but found himself on the sidelines this term and was only able to make 10 league starts.
The 33-year-old was an unused substitute during Hull's FA Cup final defeat to Arsenal last weekend, meaning his last appearance for the club was their 2-0 defeat to Everton in the last game of the league season.
Koren, who moved to the KC Stadium from West Brom in 2010, is joined on the released list by central defender Abdoulaye Faye, but goalkeeper Steve Harper will remain for another season after triggering a clause in his contract.
The top-flight club also hope to agree new contracts with forward Matty Fryatt, goalkeeper Eldin Jakupovic and defender Joe Dudgeon in the coming days.
Hull finished 16th in the Premier League this season, four points clear of relegation, and Steve Bruce's side will play in Europe next season as beaten Cup finalists.
