The long-serving Eagles winger has entered the final 12 months of his current deal at Selhurst Park and has been linked with a move to Serie A club Roma in recent weeks.
A report in 90Min has suggested Chelsea are keen to make a move for Zaha and would offer wantaway winger Callum Hudson-Odoi in a deal to take the Ivory Coast international to Stamford Bridge.
The report states Zaha is prepared to leave Palace to secure an opportunity to experience Champions League football and challenge for major trophies.
However, Eagles head coach Patrick Vieira has insisted the winger is keen to remain at the club, despite speculation linking him with a move away.
Speaking last month, he said: “We are in a period where there is a lot of speculation regarding players.
“What is really important for me and the football club is that Wilfried Zaha came back on time and he is working well to have a good season.
“What goes on in the papers, it is difficult for us to control. What I know is that he is happy in our football club and he is looking forward to the new season at our football club.”
A potential swap deal involving Hudson-Odoi would offer Palace a direct replacement for Zaha and allow Chelsea to part company with a player that has reportedly asked to leave the club over the last week.
The 21-year-old is believed to have received permission to secure a temporary move away from Stamford Bridge to satisfy his desire to find regular first-team football during the new season.
German giants Borussia Dortmund and Premier League trio Newcastle United, Leicester City and West Ham United are all said to be keen on securing a loan deal for the three-times capped England winger - but their hopes could be dashed if Chelsea’s move for Zaha comes to fruition.