Earlier this month, Goal revealed the Toffees had made the Spaniard their preferred choice to succeed David Moyes, after the Scot left to take over at Manchester United.
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"We've been friends for many years but in football there comes a time and now is the time," Whelan told reporters of Martinez.
"He feels he's not the man to lead us back 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 Latics owner, however, has warned that any club interested in taking the Spaniard will have to pay the remaining 12 months of his contract.
In four years at the DW Stadium, Martinez kept the Latics clear of Premier League relegation three times and won admirers for his expansive, attacking brand of football.
The Spaniard’s crowning glory came at Wembley earlier this month, as Wigan upset the odds to defeat a lacklustre Manchester City side and claim their first ever FA Cup.
But defeat to Arsenal three days later confirmed their return to the Championship after an eight-year stay, and Martinez has now decided the time is right to step away.