The offer for the Ivory Coast winger is cash-only and does not include the possibility of any Everton players moving in the opposite direction.
Arsenal football club had previously appeared to be favourites to sign Zaha, but the London club are now close to completing a deal for Lille winger Nicolas Pepe.
Zaha’s preference was for a move that would keep him in London, but he is understood to be willing to move back to the north west, despite an unsuccessful two-year spell at Manchester United before he returned to Palace.
The 26-year-old scored a career-best 10 Premier League goals last season to help Palace to a 12th-placed finish, but now wants to prove himself at a higher level.
Everton are also in talks to sign Juventus striker Moise Kean, with the Toffees looking set to continue the aggressive transfer policy they have pursued since majority owner Farhad Moshiri invested in the club in 2016.