Shockingly though, Zaha has told his teammates that there is "an offer for him to leave Selhurst Park", which reads as a major blow to Palace supporters.
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.
But the Mirror suggest that the Ivory Coast worldwide has rejected a fresh extension that would improve his terms even further to £125,000-a-week.
The Englishman's failed spell at Manchester United prompted a return to Palace, but Zaha was vital in his side's quest to avoid relegation last season.
Zaha is interested in leaving Palace he had even submitted a transfer request but does not want to leave London, with Chelsea being his most preferred destination.
Heading into the summer transfer window, Chelsea were in a precarious situation with Eden Hazard and Willian heavily linked with summer departures away from Stamford Bridge. Zaha is thus, a viable target, who will bolster the Blues' attack irrespective of the Brazilian's departure.
But, according to Guardian Sport, Crystal Palace are determined that they will not sell Zaha.
Jorginho and Rob Green are the Blues' only signings so far, with Sarri keen to splash the cash before the transfer window shuts on August 9.
The 25-year-old has been on Tottenham's shortlist for the last two years but Spurs have never been able to agree a deal with Palace. But, Chelsea will hope to seal the deal when they table their late offer for the Ivory Coast worldwide.