The Republic of Ireland international has just a year left on his current contract at the London Stadium and will be available for a low fee.
Cullen is not in manager David Moyes’ plans going forward, and could be willing to move to their city rivals in order to start playing Premier League football.
The 24-year-old spent last season on loan in the Championship with Charlton Athletic after helping them win promotion the year before, featuring in 34 league games and scoring once and assisting twice.
Fulham clearly was impressed with his performances and are looking to add the necessary quality to their midfield following an instant return to the top-flight.
Cullen’s expertise when it comes to set-pieces and ability to dictate the tempo of games are qualities that will come handy in the top-flight and Palace also reckon they could do with him.
The Hammers academy graduate will want the assurance of regular football before joining any EPL side, and it will be interesting to see where he is plying his trade after the end of the transfer window.