The 29-year-old has a just year left on his current Palace deal and there has not been any progress with regards to getting him sign an extension.
Earlier in the summer it was AS Roma who were interested in the former Manchester United man and even though the deal has gone quiet, Chelsea are now beginning to look at him seriously.
Thomas Tuchel’s side initially had a list of strikers they wanted, but it has been a frustrating summer so far with only the signings of Raheem Sterling and Kalidou Koulibaly serving as a source of respite.
The loss of Andreas Christensen to Barcelona and that of Antonio Rudiger to their rivals Real Madrid has left the team short of options and quality at the back.
But it is the investment on Romelu Lukaku, who has just gone back to Inter Milan on-loan after just a season at Stamford Bridge, which has necessitated the reinforcement in the attacking line.
According to a report on French website Media Foot, Tuchel is a huge admirer of Zaha and is willing to take advantage of his current contract situation at Selhurst Park to bring him to Stamford Bridge.
Last season, Zaha led from the front, playing a crucial role as the Eagles impressed under Patrick Vieira.
He scored 15 goals in all competitions and looked very happy playing on the right side of attack and occasionally through the middle.
How much will Chelsea be looking to pay him ?
The Ivory Coast striker is currently on £130,000-a-week wages at Selhurst Park, and reports are that Chelsea will have little problems in offering to bump it up to £200,000.
With the summer transfer window well and truly in swing, it will be interesting to see how this new development will affect the plans of manager Patrick Vieira and the player himself.