Arsenal and Everton have been keen on signing the 26-year-old winger this summer.
Everton saw a £70m plus James McCarthy and Cenk Tosun offer for Zaha rejected on Wednesday.
And Zaha, who was disappointed by the news, has now handed in a transfer request at the club. The Eagles are reportedly looking for at least £70m for the Ivory Coast international.
The Gunners were interested in signing Zaha but were put off by the huge price tag and went for Lille’s Nicolas Pepe instead in a £72m deal.
Meanwhile, Palace have made an improved offer worth £23m for CSKA Moscow striker Fedor Chalov. The Eagles had their initital £14m bid rejected last week.