The Times understand that Spurs failed with two offers during the 2016-17 campaign but have not been put off and will reignite their interest this summer.
Spurs move into their £750m new stadium (The Daily Mail) in the summer but the report claims the move will not hinder their plans in the transfer market.
Chelsea and City are both noted as rivals for the Ivory Coast international, but Tottenham are hopeful of securing a player they offered £21m for in 2016, the story adds.
Mauricio Pochettino's side are said to be confident that a deal can be struck to bring Zaha - who signed a new contract at the end of last season - to north London even if the Eagles are successful at avoiding relegation to the Championship.
Palace currently sit just a point above the relegation zone and face Tottenham this weekend at Selhurst Park in the first of three scheduled meetings with sides who occupy the current top six in their last 11 matches.
Zaha's form has been key to keeping Palace's heads above water with four goals and two assists in 20 Premier League appearances but co-owner Steve Parish is unwilling to allow the wideman to depart without a fight.
"He will certainly be staying at the club in the foreseeable future," Parish told the BBC World Service in January, according to the Croydon Advertiser.
"I can't say forever, he has a contract that eventually will run out and we all know that we are free people. Right now, Wilfried [Zaha] is at the club, he wants to be at the club and we want him to be at the club and we all believe he is at the right place. He believes he is at the right place, he has said it, we have said it.
"Right now, we are focused on the next 14 games and what we need to achieve in the next 14 games, and how we achieve that. And then we will sit down with the manager and the sporting director and look at what we need to do to grow and improve the club."
As well as the opportunity to play in a new 61,000-seater stadium and for a side regularly contesting for the Premier League title, a move to Tottenham would potentially offer Zaha a chance to make his debut in the Champions League – a competition in which he was not afforded an appearance during his brief spell with Manchester United.