Martinez has struggled all season at Goodison Park, following a similarly disappointing campaign last term. Everton's poor Premier League form had been excused to a degree thanks to their long FA Cup run.
Saturday's semi-final defeat to Manchester United, however, has put paid to any hopes of silverware. Another season without a trophy and Everton sitting in 11th place doesn't exactly fill the Toffees with confidence.
That's why speculation is growing about Martinez's future, with the Merseyside outfit reportedly having drawn up a shortlist to replace him.
On that shortlist sits Southampton head coach Koeman. But how do Everton and Southampton fans feel about the prospects of the Dutchman switching clubs?
Those at Goodison Park are excited by the idea of Koeman replacing Martinez. But those at St Mary's don't think he'd take the 'step down'...
Yeah, Koeman's bound to leave #SaintsFC for a club who are finishing below us for the second season running.— Richard Emery (@UrbanWordsmith) April 24, 2016
People wanting Koeman,realistically why would he leave Southampton for Everton ??— Bradlottie (@Bradlottie1) April 24, 2016
I wouldn't mind seeing Koeman as our new manager. He's done well at Southampton after everyone doubting him. #EFC— Daz Shire (@BluDazza70) April 24, 2016
Really want Everton to go for Ronald Koeman. Mourinho is a step to far for me, he wil end up at 1 of the best. Koeman is a perfect fit #efc— Mark O'Malley (@MalleyNSNO) April 24, 2016