He has already snubbed interest from Everton as he is reluctant to move north again, after a failed stint with Manchester United.
The Ivorian winger struggled to make an impact during his time at Manchester United but a move across London may well suit a player who many believe is the best in the business in the Premier League outside of the top six clubs.
Tottenham have also been linked with Zaha, but are now trying to sign Aston Villa's Jack Grealish.
Blues boss Maurizio Sarri is now facing an attacking headache with Brazilian worldwide Willian set for a big money move to Manchester United or Barcelona, and Eden Hazard looking to force a move to Real Madrid.
The same article states that the Blues are willing to use England worldwide Danny Drinkwater in an attempt to sign Zaha to improve their attack.
Should Chelsea land Zaha, he will become Maurizio Sarri's third summer signing.
But, Chelsea are hopeful of finally landing the 25-year-old with an improved offer worth £45 million, plus £15 million in add-ons and a massive sell-on fee.
But Allardyce, Palace manager during the 2016-17 Premier League season, thinks the time is right for the increasingly versatile Ivory Coast global to move on from Selhurst Park once more.
That deal would have increased his wage to £125,000 per week, but Zaha is well aware that he could earn significantly more at Stamford Bridge. An interesting next few days beckon.