The Daily Mirror says the Blues will launch a bid for Zaha in the summer if Palace refuse to sell the Ivory Coast international during the January transfer window.
Arsenal have also been linked with a £35m ($47m) move for the 25-year-old as a replacement for Alexis Sanchez, who is in the final year of his Gunners contract and is expected to leave the north London club on a free transfer in the summer.
Zaha has scored four goals in 12 Premier League appearances for the Eagles this season. He signed a five-year contract extension at Selhurst Park in May.
The winger left Palace to join Manchester United in 2013 but rejoined them two years later after two loan spells at the south London club.
Chelsea could face competition for Zaha from Tottenham, whose manager Mauricio Pochettino is understood to be an admirer of the winger. Manchester City were also believed to have made contact with his representatives during the summer.
Palace chairman Steve Parish said last month there was no chance of Zaha being sold in the January transfer window.
"That's out of the question [him leaving]," Parish told BBC's Match of the Day. "We need to keep going, make sure we get as high as we can up this league.
"Working with Wilf and having Wilf around the club is fantastic. He cares so much about this club.
"He's a joy to work with for every manager, and obviously he makes a big difference to us on the pitch."
Palace are in 14th place on the Premier League table, two points clear of the relegation zone.
The Eagles' next league assignment sees them taking on Swansea City at the Liberty Stadium on 23 December.