The Spaniard led Everton to a fifth-placed Premier League finish in his first season in 2013-14, but they struggled last term with the added workload of playing in the Europa League.
This season the club lie 12th with only six league wins in 23 matches and their chance of ending a 21-year wait for a trophy in the League Cup died last night as they threw away a two-goal tie lead against Manchester City.
Simon Magner, chair of the Everton Supporters Trust, told PA: "The feeling among fans from what I've seen over the last couple of weeks is that Martinez has been more and more running out of time with the fanbase. Certainly, for a lot of fans, the semi-final was his last-chance saloon.
"I have never known as many people as there are now admitting the time has passed for the manager and it is time for him to move on.
"I think it is safe to say the vast majority of the fans are thinking time is up now and that it is time we tried something else."
Everton are still in the FA Cup and travel to Carlisle United in the fourth round on Sunday.
See what the Sports Mole team had to say about Martinez's future in the latest episode of The Dugout.