David Moyes will be sacked as Manchester United manager, according to reports.
The loss at Goodison Park was United's 11th of a wretched title defence and killed off any last hopes of a return to the Champions League next season with the gap to fourth-placed Arsenal now too big to bridge.
United were knocked out of this season's Champions League by holders Bayern Munich in the last eight, having led briefly in both legs.
They were eliminated from the FA Cup at the first hurdle by Swansea and were knocked out of the Capital One Cup at the semi-final stage by Sunderland.
Moyes continues to paint a positive picture about the future, but he cannot hide from the statistics that show just how badly his first season at United has gone.
United will end the season with their worst points total in Premier League history. They have failed to qualify for the Champions League for the first time in two decades.
And they will only guarantee Europa League football at Old Trafford next term if they overhaul the six-point gap that separates them from Tottenham.
Moyes, speaking to MUTV after the defeat at Goodison, is hoping to end the season on a high with wins in United's remaining four games, but the reports on Monday suggest he may not be afforded the opportunity to even see out the campaign as United boss.
''We need to end the season on a high,'' the United manager said.
''We want to finish by winning all of our games. We'll do everything we possibly can to make that happen.''
Yet another blow hit Moyes on Sunday morning when it emerged that forward Danny Welbeck was considering his future at the club following a lack of game time this year.
The embarrassment among the playing staff at losing yet another game so easily was clear on Monday morning.
Rio Ferdinand, who was not in the squad at Goodison Park, took to Twitter to air his feelings.
''Frustration & disappointment sums up the season so far,'' the United defender wrote.
Juan Mata had started to pay back some of his £37.5million price tag in recent weeks, but the Spaniard had an off-day at Goodison Park.
''I feel very disappointed today,'' United's club-record signing wrote on his personal blog.
''This is how you feel when you spend two weeks waiting for a game, getting ready for it, and then things don't go the way you wanted. It happened in Goodison Park and the truth is this is not the first time, as you know.
''I hate this feeling.
''What the Premiership table shows now is something that neither the club nor the fans deserve.
''I hope the next game comes as soon as possible because, as you can imagine, this bad taste makes me mad.''
Moyes gave a damning critique of his team's defending for both goals, describing it as ''rank'' and ''rotten''.
Leighton Baines converted a penalty to give Everton the lead when Phil Jones inexplicably stuck his arm out to block Romelu Lukaku's shot before Kevin Mirallas slipped through the away back line and beat David de Gea from a tight angle.
Wayne Rooney, captain on his return to Goodison Park, said recent defensive performances have not been up to scratch.
''We conceded two really sloppy goals, which we need to cut out because we're making it hard for ourselves to get back into games,'' the United striker said.
''For the second one, we should have stopped the counter-attack. They ran in behind us. It's not good enough.''