The Toffees have already strengthened this summer with the arrivals of Richarlison and Lucas Digne.
Brazil attacker Bernard has completed a medical at Goodison Park.
The defender has been heavily linked with a move to Everton but faced late competition from Manchester United.
Everton will be keen to give Gomes a chance to remind his form, with the talented midfielder left out of the Portugal squad for the World Cup this summer. My understanding, I've spoken to somebody in Spain this morning, is that one is very, very close to being done.
England v India: Rain delays start of second Test at Lord's
India struggled against the swinging ball, with England finding it hard to combat Ravichandran Ashwin's off-spin. Neither side made 300 in the match, England posting 287 before India replied with 274 in the first innings.
It follows a window which saw them offer under the odds for Harry Maguire and Jerome Boateng, about whose fitness they had reservations, while they refused to meet the demands of Yerry Mina's agent and didn't even register an offer for Spurs' Alderweireld.
The Everton manager also let Ashley Williams leave the side on loan to join Stoke City and Ramiro Funes Mori was allowed to join Villarreal.
Usually in cases such as this clubs can approach the Premier League and request more time to complete the deal, if they can provide all the necessary paperwork and prove that the completing the deal is just a matter of dotting i's and crossing t's. There's been plenty of names, clubs who have been after him.
However, Chelsea did not want to sell the player to a direct rival and Everton appear to have won the race to sign the player.