The Toffees have been active in the transfer window over the last couple of weeks bringing in both James Tarkowski and Dwight McNeil, who were teammates at Burnley.
Frank Lampard has also moved to secure the services of former Wolves full-back Ruben Vinagre on-loan from Sporting Lisbon.
He is not stopping there, though, and is now keen to add to his forward options with a move for Chelsea outcast Batshuayi.
The Belgium international is seen as a replacement for Richarlison, who left to join Tottenham in a £60million deal.
Batshuayi moved to Stamford Bridge from French side Marseille in 2016 but has spent most of his time with the London club out on loan.
He has had spells across Europe with the likes of Borussia Dortmund, Valencia, Crystal Palace and, most recently, Besiktas, where he netted 14 times in 42 appearances in all competitions.
The 28-year-old is now expected to leave Stamford Bridge on a permanent basis if Everton are able to strike a deal for the player.
Batshuayi is also entering the final year of his contract with the Blues, and Thomas Tuchel is prepared to offload the forward, who is not in his plans for the upcoming season.
Everton are also understood to be closing on the signing of for former midfielder Idrissa Gueye, who has fallen down the pecking order at Paris Saint-Germain.
talkSPORT understands the Senegal international is set to have his Goodison Park medical.
Ross Barkley is another name to have been linked with a return to his boyhood club in recent weeks.
The Toffees have also previously expressed their interest in a host of other Chelsea stars, with Conor Gallagher, Billy Gilmour and Armando Broja all having been linked to move to Goodison Park.