Everton went down 4-0 to Manchester United after a flurry of late goals.
Henrikh Mkhitaryan, Romelu Lukaku and Anthony Martial scored in the final 10 minutes after Antonio Valencia’s thunderbolt.
And Ladbrokes have slashed the odds on Koeman getting the sack to just 15/8.
Koeman oversaw a big overhaul at Everton this summer but they have struggled in Lukaku’s absence.
Everton spent big on Jordan Pickford, Michael Keane and Gylfi Sigurdsson.
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But they dropped into the bottom three this afternoon after their six games in all competitions without a win.
“Sometimes you get more than you deserve,” he said.
“We made a big mistake at 1-0 and after that it was over. From what I saw today from the team it made me happy. More happy than Thursday.
“We grew in the game and in confidence and not many teams will get two big chances like we got today. We didn’t score the goal and after a personal mistake it was 2-0.”
Everton have shipped 10 goals in three games – losing 3-0 to both Atalanta and Tottenham ahead of United.
Prior to that they lost 2-0 to Chelsea and drew with Hajduk Split and Manchester City.
Everton drop into the bottom three on goal difference – which is now minus 8 – going below West Ham.
Things may get easier for Koeman’s men with Sunderland in the EFL Cup on Tuesday before facing second-bottom Bournemouth.