Gomes, also linked with Premier League rivals Tottenham and West Ham, has signed a five-year contract at Goodison Park.
The Portugal midfielder appeared 29 times for Everton in all competitions last season after joining on a deadline-day loan in August 2018.
"I'm really happy to sign the contract with Everton - it wasn’t hard to decide, it was an easy decision and I’m very happy to have made it," Gomes told his new side's official website.
"I know the club made a massive effort to sign me and I'm very happy and thankful to them. I said during the year that I felt part of a family and that is the most important thing to me.
"Last year was a really good experience for me. I just wanted to feel part of something special and I found it here. It was good for me in that moment and right now, after signing for Everton, it's even better.
🗣 | "It wasn't hard to decide. We are all excited for the future." - @aftgomes.— Everton (@Everton) June 25, 2019
"I think the ambition here is not only for me, I think it's for all the players, the fans and the club. We are all excited for that."
Valencia appeared to confirm the arrival of Jasper Cillessen from Barcelona on Tuesday, meanwhile, in a deal that is expected to see fellow goalkeeper Neto head to Camp Nou.
The Copa del Rey winners, who beat Barca in the final, posted a video on social media seemingly showing Netherlands international Cillessen collecting a pair of orange clogs bearing the Valencia badge.
Barcelona's clear out continued with Marc Cardona being given the green light to join Osasuna in a deal worth €2.5m.