The Daily Star says Zaha has been a long-term target for Liverpool, who are keen to continue their bid to recruit top English talent.
Moneybags Manchester City are also keen.
Both clubs sent scouts to Selhurst Park to watch Zaha star in Palace's Carling Cup win over Championship high-fliers Middlesbrough last week.
Meanwhile, Stoke and Fulham, who had approaches for Zaha knocked back during the summer, remain interested.
Zaha penned a four-and-a- half year deal last November tying him to Palace until 2015.
But the club appear resigned to losing Zaha, their young player of the year last season, well before then.