Puncheon has been a Palace regular throughout his time at the club - until last season, that is, when he started just six Premier League games before his campaign was cut short by a season-ending injury in January.
The Sun claim Middlesbrough are also interested in Puncheon but he has rejected the chance to link up with his former manager Tony Pulis at the Riverside.
Any move for Puncheon could depend on whether Yannick Bolasie returns to Palace. The Eagles have been linked with a move for their former winger and his arrival would open the door for Puncheon to leave.
It has been an infuriating transfer window for Newcastle fans, who have been given a feeling of déjà vu as chairman Mike Ashley has once again refused to back popular Magpies boss Rafa Benitez in the transfer window.
Newcastle needed to spend big this summer but in fact they have made a £22.9m profit, leading fans to question where the money from player sales is going.
Ashley promised Benitez "every penny generated by the club" in an effort to keep him at the club earlier this summer, but he has gone back on his promise and Benitez has not disguised his anger in recent press interviews.
Newcastle's second season back in the Premier League kicks off against Tottenham on Saturday.