Tottenham are also interested in the Ivory Coast global.
A five-year contract extension last year tied him up to the summer of 2022, but a big bid is tipped to be enough to lure Zaha in and the Blues, struggling to retain their stars as well as failing to land preferred transfer targets in absence of Champions League football, will have a significant boost if they manage to strike a deal before the new Premier League season kicks off next weekend.
Zaha has found his mojo once again since returning to Crystal Palace from Manchester United in the summer of 2015.
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Everton turned to Watford wideman Richarlison instead, and now Zaha is up for a move across London to Stamford Bridge. Zaha is thus, a viable target, who will bolster the Blues' attack irrespective of the Brazilian's departure.
That gives Chelsea a clear run at the player, although significant hurdles remain if they are to agree a deal before the transfer window closes.
Tottenham have long held an interest in the talented winger, while Chelsea have also emerged as a potential destination for the player.
Ivory Coast worldwide Zaha impressed with nine goals and made five assists in 29 games in the Premier League last season to earn rave reviews for his performances.
Palace will not sell the Ivory Coast worldwide unless they are offered a huge sum to sell him for a second time in his career. An interesting next few days beckon.