The Magpies expressed disappointment at the striker's comments but stressed that the 25-year-old remains under contract and will be welcomed back into the squad.
"We are disappointed to have been alerted to a social media post by Alexander Isak this evening," read the statement released late on Tuesday.
"We are clear in response that Alex remains under contract and that no commitment has ever been made by a club official that Alex can leave Newcastle this summer.
"We want to keep our best players, but we also understand players have their own wishes, and we listen to their views. This is a proud football club with proud traditions and we strive to retain our family feel.
"Alex remains part of our family and will be welcomed back when he is ready to rejoin his teammates."
Since joining from Real Sociedad in 2022, Isak has scored 62 goals in 109 appearances.