According to the Evening Chronicle and the Daily Mirror, both clubs had wanted to sign the England youth International this month.
A January move for Zaha could be put on hold due to Palace being just 90 minutes away from a place at Wembley.
The South Londoners secured a 1-0 first leg victory over Cardiff and could face either Liverpool or Manchester City in the final of the Carling Cup.
A trip to Wembley would create a cash windfall for Palace, who were in administration just two years ago, meaning they won't need to sell their star asset until the summer.
Source: DSG