Alongside head coach Unai Emery, himself linked with Goodison but also with roles AC Milan and PSG, the 47-year-old former goalkeeper helped the Spanish club to their third successive Europa League triumph last month which has elevated his stock in the game even further.
Regarded as one of the best technical directors around, Monchi surprised Sevilla last week by requesting to leave the club despite having four years left to run on his contract.
The timing coincided with escalating speculation regarding Emery and Everton, with many media sources suggesting that Farhad Moshiri wanted to bring the pair in to replace Roberto Martinez in a revamped managerial structure at Finch Farm.
Monchi has also been talked about in terms of fulfilling the same role regardless of who is appointed as Martinez's successor but his latest comments suggest he is looking to take time out from football.
“I asked them to let me leave, because I'm aware of the terms of my contract. I needed to rest. Simply that," Monchi is quoted by AS as telling Seville radio. "I wasn't aggressive because I had offers, but because I know Monchi better than anyone. If it's not immediately resolved then I don't get what I wanted, which is rest.
“There is a contract and a clause. I wish they had appreciated my need but they have done what they had to do. When I asked to leave it is the first time in 28 years I have put Monchi ahead of Sevilla. I won't pay the clause, no. I couldn't leave by the back door.
He seems prepared to abide by Sevilla's decision not to release him from his contractual obligations and it doesn't look as though he will be leaving the club he has served for so many years any time soon.
“If anything I'm going to deliver more than ever," he continued, "because I owe the fans that effort. My greatest joy is to make the fans happy and that will again be the target.
“I was asked if I saw another solution [than leaving] and the only one I could see was to get a new brain. I'll try to change things without slowing down in my way of working. That I can't change, but there are things we will try together. I won't lift a millimetre off the throttle.”