Sevilla president Jose Castro does not believe Unai Emery will leave for Everton this summer, but Frank de Boer remains keen on the job at Goodison Park.
Everton are seeking a replacement for Roberto Martinez this summer and club sources have told ESPN FC they are assessing a number of options, including Southampton's Ronald Koeman and Manuel Pellegrini, while David Moyes has said he would consider a return.
Sources also told ESPN FC that Everton were due to meet with Emery this week, but the Spaniard has said he is "very well" at Sevilla and Castro expects him to remain with the Europa League winners.
"The coach has understood our idiosyncrasies perfectly," he told Sevilla's TV's "A Balon Parado" programme. "He is crazy about football, the hours he puts in, how he manages the squad. I say that now after being successful, but also at the start of the season.
"I can tell you he has said he is not on the market and is not thinking about anything else."
De Boer, who left Ajax this month, is also among the potential options and has previously indicated an interest.
His brother, Ronald, told the Liverpool Echo on Tuesday: "Frank is keen. If he has a chance to be in the Premier League then he will want to take it.
"Everton are a big club and it would be a challenge, but I know for sure he would take the job if offered. But that's up to the management at Everton. He is probably on their list.
"He has to wait and see what the management does at the club, but I hope he will get the job."
He added: "He hasn't spoken to Everton."
ESPN FC's Spain correspondent, Dermot Corrigan, contributed to this report.