A heavy setback at Leicester City on Sunday could persuade owner Farhad Moshiri to act immediately and sack the Portuguese.
Pickford, 25, has already played under four different managers since signing for Ronald Koeman at Goodison in the summer of 2017 and it'll quickly turn into five unless results improve immediately.
"It was obviously deflating after the Norwich game. Everyone was down and it wasn't nice," acknowledges Pickford.
"No one likes getting beat. Normally I go and shake the other team's hands and say well done but this time I walked straight down the tunnel.
"I wasn't fuming, I was just disappointed we got beat. We need to start picking up results. I don't want to see the manager sacked.
"I was here when Koeman lost his job and It wasn't nice to see at all, it was horrible. We were in the building when it happened, and you never want to see anyone lose their job like that.
"The manager has been great with us and we are behind him 100 per cent. A few bad performances shouldn't put the manager under pressure but that's the industry we are in. Unai Emery has just lost his job after a bad run of form. You want stability throughout a club, don't you?"
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