Thomas Tuchel is eager to bolster his attacking line ahead of the transfer deadline.
The Blues have allowed striking pair Romelu Lukaku and Timo Werner leave for former clubs Inter Milan and Red Bull Leipzig respectively.
Chelsea have been trying to strike a deal with Barcelona for Pierre-Emerick Aubameyang, but have been unable to for weeks now.
With time running out before the close of the deadline, Tuchel is now keen to lure Zaha to Stamford Bridge.
Crystal Palace have so far resisted selling the Ivory Coast international to a Premier League rival, previously placing an £80milllion price tag on the 29-year-old.
However, as Zaha now has less than one year left on his contract, Palace may decide to cash in on the winger now rather than allow him to depart on a free transfer next summer.