Oumar Niasse of Everton (left)
Niasse has been strongly linked with a move to the Seagulls, with The Sun reporting last week that a deal 'will be agreed' to take the Senegal striker to the Amex Stadium.
Brighton manager Chris Hughton
But according to The Guardian, Palace are also in the hunt for the player, although unlike their fierce rivals it is claimed they have yet to make an official offer.
Everton are said to be demanding a £10 million fee for Niasse, however, the report states that both of Brighton's bids are loans with an option to buy at the end of the season - the latest involving a £750,000 loan fee.
The 27-year-old has yet to feature in a competitive Everton fixture during Ronald Koeman's tenure as manager, and spent the second half of last season on loan to Hull City.
Oumar Niasse of Hull City
According to Transfermarkt, the Toffees paid around £16 million to sign the former Lokomotiv Moscow player 18 months ago, but he has failed to score in seven Toffees outings.
Both Palace and Brighton are down to one fit striker - Christian Benteke and Tomer Hemed respectively - which could play into Everton's hands.
Could Crystal Palace beat Brighton to Niasse's signing?