However, there was unrest among a lot of fans during tonight's 1-1 FA Cup draw with West Ham at Goodison Park and only a last-gasp goal from Romelu Lukaku averted a fifth straight defeat.
And some fans are talking - some hypothetically, some not - about who they would like in if Martinez does depart.
How about an Everton legend; who has been a fan as man and boy; who has coached at the club before and holds a Uefa Pro Licence; who was interviewed for the job in 2013; and who is already impressing in his first stint in management?
Step forward Alan Stubbs. Yet I don't hear his name mentioned very often.
Granted, he's only half a season into his first job in management, at Hibernian, and he had a tough start to life in Scotland.
When he arrived the squad was threadbare, demoralised and there were literally no backroom staff.
And results were disappointing to start with, with city rivals Hearts taking a massive lead in the Scottish Championship, which they still hold.
Yet Stubbs revamped the backroom staff, introduced a sports science department and introduced an attractive style of football. And it finally seems to be working.
Hibs thumped Rangers 4-0 during the festive period and drew away at arch rivals Hearts. In fact they have been on an impressive run of form for the last couple of months.
I spoke to Stubbs, 43, a few weeks ago, and he admitted Everton was his dream job.
"One day it would be really good to go back there," he told me. "I’ve thought about it.
"Could it ruin relationships there and potentially tarnish my relationship with the fans? Because I am held in high esteem by them.
"But to say you have played for, and managed, the club you supported as a kid – the club you still support - that would be quite something. I’d be very proud of that."
Stubbs enjoyed cult status during two stints as an Everton player, from 2001-2005 and 2006-2008. Then, in 2008, he was asked, out of the blue, by David Moyes to join his coaching staff.
Moyes was - and is - a mentor to Stubbs, although he emphasises that he wants to remain his own man.
Stubbs was very happy leading the under-21s at Everton, but his ambition led him north of the border.
"The easy thing would have been to stay at Everton and be under-21 coach and have no pressure or worries," he admits.
"As a kid, my ambition was to be a professional footballer, and play for my own team, and I did that. Now my ambition is to manage at the highest level."
Could he reach that highest level with his boyhood club?