Two of Big Sam’s former clubs, Palace and Everton, have been locked in discussions over a deal for the Ivory Coast forward over the past week.
It is claimed Everton’s latest offer amounted to around £52m-£55m up front.
This was more along the lines of Palace’s demands, having rejected previous offers from the Toffees and Arsenal which featured staggered payments, but it still fell short of the club’s estimated £80m price tag.
Now reports claim the Eagles have told interested clubs Zaha is no longer for sale and will stay at Selhurst Park this summer.
But, with less than two weeks of the transfer window remaining, Allardyce believes a deal could still be done and Palace could be tempted to sell their talisman if the price is right.