The 47-year-old has been a longstanding servant of the Liga club and is credited as being the brains behind Los Rojiblancos' impressive Europa League treble but he rocked the club last month by asking to step down from his role.
Initially it was thought he was quoted as saying he was looking for a new challenge but he has since described his desire to take some time out form the game to rest and be with his family.
Sevilla's refusal to release him from his contract four years early means that he looks likely to remain in his post but, assuming the Echo's report holds water, Everton could trigger a £3.8m release clause to get him on board as part of Farhad Moshiri's plans to restructure the team management and player recruitment set-up at Everton.
The Blues would face competition from Paris St Germain if they are successful in hiring Unai Emery, however. The French champions are reportedly in talks with Emery, the man who has led Sevilla to three successive Europa League titles alongside Monchi, about the possibility of succeeding Laurent Blanc as their manager.
Blanc led the moneybags Parisien club to a runaway title in Ligue 1 but failed to negotiate their way past the last eight of the Champions League which is the ultimate goal for PSG's Qatari owners.
Assuming Monchi is looking for a new role at the moment, a crack at continuing his relationship with Emery at one of Europe's richest clubs would no doubt appeal but he has also expressed an interest in working in the Premier League.