The 20-year-old has already admitted he wants to make the step up to the Premier League in the New Year, if his current club agree to a transfer.
Gunners boss Arsene Wenger is desperate to add to his Emirates Stadium squad in the forthcoming window, particularly after Manchester United ruled out selling Nani to their rivals.
Palace chairman Steve Parish insists Zaha will not leave for less than £20million, but it is believed a bid of half that will test the Championship club's resolve, and see how determined they are to keep the recently capped England international at Selhurst Park.
"Playing for England. So what? It's four minutes," said Eagles boss Ian Holloway, who wants Zaha to concentrate on improving his form for his current club, rather than setting his sights on a move.
"I don't want to sound funny but I'm more concerned about us working with him and getting him playing how he was a couple of weeks ago. That's all I worry about."
Zaha is a self-confessed Arsenal fan which should give Wenger an advantage over Manchester United in the race to seal the youngster's signature, but Sir Alex Ferguson is unlikely to give up easily on his attempts to lure the talent to Old Trafford.