Press Association Sport says Spurs are unlikely to make an improved bid for the former Manchester United player and have moved on to other targets.
Zaha appeared close to an exit from Selhurst Park after telling Palace boss Alan Pardew on 25 August that he wanted to leave the club.
But with no other offers on the table, the 23-year-old looks set to stay with the Eagles.
Palace chairman Steve Parish criticised Tottenham last week for trying to turn Zaha's ahead with a "ridiculous" offer.
"We had a bid from Spurs of £12m – it's a ridiculous bid. I can't imagine that they're really serious," he told talkSPORT.
"There is no chance whatsoever of Wilfried Zaha leaving the club in this transfer window.
"I can absolutely reassure every Palace fan he is a massive part of what we want to do.
"We've told Spurs that there is really no point bidding for him because he is going nowhere – 100% guaranteed."
'Need Mane-like signing'
Zaha has played in all three of Palace's league games this season. He started 30 games for the Eagles last campaign, scoring two goals.
Speaking after Tottenham's 1-1 draw with Liverpool last weekend, Spurs boss Mauricio Pochettino said he was looking to sign a player in the mould of new Reds recruit Sadio Mane.
"We need a player that is more direct, a player that is more aggressive and offensive," he was quoted as saying by the Liverpool Echo.
"We have a player like [Christian] Eriksen or Son [Heung-min] or [Erik] Lamela that are players who like to play into feet.
"Some characteristics like that we see today from Liverpool. Mane, this type of player that can be offensive."
Spurs are close to completing an £11m deal for Marseille winger Georges-Kevin Nkoudou.