Arsenal are the reported frontrunners for Zaha as the Gunners have been linked with a £35m ($47m) move for the 25-year-old as a potential replacement for Alexis Sanchez.
According to The Mirror, City have also begun the groundwork on a potential deal for Zaha.
Pep Guardiola's side are the favourites to snatch Sanchez away from Arsenal, with just six months left on the Chilean's current deal and having nearly finalised a move to the Etihad Stadium on transfer deadline day this past summer.
However, the Citizens view Zaha as an alternative to Sanchez with Paris Saint-Germain expected to provide stiff competition for the 28-year-old's signature. Landing the Abidjan native also allows Guardiola to keep his current attacking options well-stocked.
Palace chairman Steve Parish though, claims there is no chance of Zaha being sold in the January transfer window.
"That's out of the question [him leaving]," Parish said in November. "We need to keep going, make sure we get as high as we can up this league.
"Working with Wilf and having Wilf around the club is fantastic. He cares so much about this club. He's a joy to work with for every manager, and obviously he makes a big difference to us on the pitch."
Zaha initially made the move to a big club when he joined Manchester United from Palace in 2013 as Sir Alex Ferguson's last signing. However, he was unable to get game time and did not make a single start for the club as he would rejoin Palace in 2015 after various loan spells.
He had his best season in 2016/17 as his seven goals and 11 assists helped the Selhurst club to a 14th place finish. This season, he currently has four goals in 12 games after missing the first two months through injury.