Martinez decides to leave Wigan Athletic
The Spaniard had been in charge since 2009, having previously helped turn Swansea City into one of Britain's most attractive sides.
Wigan pulled off a series of relegation escapes under Martinez while winning admirers for their attacking football and he turned down an approach from Aston Villa two years ago.
The underdogs then managed a 1-0 FA Cup final triumph over Manchester City earlier this month to win their first major trophy but were relegated three days later after eight years in the top flight.
"He wanted permission to move so I've given that. He has 12 months left on his contract, so whoever comes in for him must pay the remainder of his contract," Wigan owner Dave Whelan told the BBC.
"He feels he's not the man to lead us into the Premier League. I've got to accept what he feels. He loves Wigan and the fans. We have to move on now and want applications for the job."
The 39-year-old Martinez is wanted by Everton, who have lost manager David Moyes to champions Manchester United after Alex Ferguson's retirement.
"I did tell Roberto before I spoke to him this morning that Everton chairman Bill Kenwright had rung me on Friday and asked for permission to speak to him," Whelan added.
"He is a must for Everton, he is a fabulous manager. In my mind he is tailor-made for Everton. I am sure Bill will be very interested in him."
Everton finished sixth in the Premier League last season, 10 points off a Champions League playoff spot.
Stoke City are another English top flight club without a coach after parting company with Tony Pulis. Much-travelled Mark Hughes has been linked in the media with the Potters.
Former Wigan midfielder Martinez, who made his name as a player in the English lower divisions rather than Spain, could also be a target in his homeland.
Reports say Malaga are set to be without a coach with Manuel Pellegrini heading to managerless Manchester City.
Martinez recruited heavily from Central America during his time at Wigan on a relatively limited budget but was criticised by fans for not shoring up a leaky defence.