According to Rob Draper of the Daily Mail, a deal could be completed within a week as Spurs look to make their first signing of the window. In regards to a fee, the Sun are claiming Palace want around £60m for the Ivorian, who has been offered a new contract amid rumours of an exit.
Mauricio Pochettino has been an admirer of the 25-year-old and hopes to see him become a Spurs player before the end of the window on August 9.
Tottenham fans have grown frustrated in recent weeks as they've seen close rivals make acquisitions, whilst their club appeared to remain silent. A deal for Zaha - for a big fee - would be sure to silence any signs of discontent from the Spurs faithful.
It would also show intent from Daniel Levy which Spurs require to push on, as they prepare to enter their new home.
After struggling at Manchester United in 2013, Zaha was loaned out to Cardiff City before returning to Selhurst Park a year later, where he has remained ever since.
Zaha has netted 22 times for Palace in the Premier League, assisting 23 goals. Playing predominantly from the right wing, Zaha has shown his capability to fill a central role which helped his side to avoid relegation last season.
Spurs have also been heavily linked to Aston Villa midfielder Jack Grealish. Tottenham are expected to pursue Grealish, who is rumoured to desire an exit from Villa Park as the Villans battle financial troubles.