Cole made the comments during West Ham’s Premier League game against Newcastle United at home on Monday evening.
The France international attacking midfielder was the star of the show at the Boleyn Ground, as he scored both the goals.
The 28-year-old opened the scoring after just nine minutes when he curled the ball past Tim Krul from the edge of the area after Mark Noble's low cross from the left wing.
The former Marseille star added a second just three minutes into the second half when he slammed the home ball from inside the Newcastle penalty box after Victor Moses had hit the crossbar following a fine run.
Payet has been superb for West Ham so far this season following his move from Ligue 1 outfit Marseille in the summer transfer window.
The former Lille attacker has scored three goals and created 16 chances in five Premier League matches for the Hammers.
The Frenchman has a pass accuracy of 81% and a shot accuracy of 75%, and has won one header, 12 dribbles and five tackles.
Former Newcastle striker Cole has suggested that Payet could leave West Ham next summer because he is so good.
"Payet looking like an absolute player.. Carries on like this will be off again next summer! " http://t.co/F8baGcyQtO— Andrew Cole (@vancole9) September 14, 2015
Cole had harsh words for his former club Newcastle and wrote on Breathesport.com: “Newcastle were very disappointing and have a very long season ahead. Yet again no goals and, to be frank, they don't look like scoring at the moment. It must be a big concern for Steve McClaren.”
The former England international added: “After a summer spending spree there doesn't seem to be any change up at #NUFC..
“With a team full of expensive foreign talent do they really know what the shirt means to the fans in the north east!”
The defeat to West Ham has left Newcastle at the foot of the Premier League table with just two points from five matches.