The Sweden international is a major target for Liverpool and Barcelona this summer.
However, The Times says Newcastle hope to convince Isak to stay by tabling a new club record contract offer to Isak's camp.
The striker is currently on £150,000-a-week and Newcastle are prepared to well beyond that figure to persuade him to stay.
Toon manager Eddie Howe has said of Isak's situation: "I never make those guarantees about any player and it’s not to do with Alex’s future but I’d be foolish to sit here and do anything like that.
“We want to keep our strongest players, I’ve said that ever since I’ve been here and sat in this seat. We have to try and keep our best players and bring in more good players to make us even stronger because the ambition of the club and everyone connected with the club is long-term success and that’s the only way to get that."