The Eagles are working on how the deal will be structured and are yet to decide about taking Oumar Niasse on loan.
The Blues made a surprise bid for the Palace winger on Monday, with the Guardian reporting that they saw an £18 million bid rejected.
They returned with an improved offer on Tuesday though, with it thought the move could cost up to £28 million with add ons.
That price would match what the Toffees paid Chelsea for Romelu Lukaku, their most expensive signing to date, and according to the Guardian the two clubs are still negotiating how the deal would be structured.
A big part of the deal is likely to be how much money Palace will receive up front, with a move for Liverpool striker Christian Benteke on the cards.
Palace are also yet to decide whether to take Everton flop Oumar Niasse on loan as part of the deal.
Bolasie is understood to be keen on the move as he fancies the chance to test himself at another club.
The newspaper also reports that he is a long-term target for Director of Football Steve Walsh, who was targeting the player when he was still at Premier League champions Leicester City.