The Toffees rounded off their disappointing 2017/18 campaign with yet another lacklustre performance on the road as a brace from Manuel Lanzini either side of half-time and a stunning strike from Marko Arnautovic were enough to overcome Oumar Niasse's sole finish.
And following the defeat - the Blues' 15th of the year - Allardyce admitted to the Liverpool Echo that he was unsure whether his future was on Merseyside.
Asked if he was confident of seeing out his contract with the club, he said: "Confident? I can't quite say after all the rumours I'm reading in the paper because generally there's something and there's no smoke without fire but I'll wait and see when we meet up next week, or this week actually."
"I wouldn't be telling you if I did", he added when questioned if he knew when his meeting with Everton's owners was due to take place.
However, Allardyce's short and spiky press conference took another turn as the former England international boss offered his curious viewpoint as to who wanted the points more.
"Did they want it more? Who had the better chances? I thought we did," the 63-year-old claimed. "I thought Adrian saved them.
"I think because Adrian was so good today with the clearcut chances that we had that he saved, we could have easily acquired a draw. But I think the quality of West Ham's goals, especially the two in the second half, was ultimately the difference - and Adrian by the saves that he made.
"We changed our system at half-time and got control of the game better, got more opportunities to attack West Ham but didn't take advantage of it and then they punished us with two very very good goals."
Niasse grabbed Everton's consolation with a well-taken finish, and Allardyce added: "When we do a little rotation of the system and the subs you see the impact they have.
"We made one at half-time and then when Theo (Walcott) came on, he and Oumar linked up better than he and Cenk (Tosun) had and I think that got us opportunities. Oumar in the box is very sharp, and that's what he got his goal from, and Theo had a goal disallowed which was marginally offside and very unlucky.
"Then Adrian saved from him, and Adrian saved again off Davy Klaassen which might have made it a very tight and close game but unfortunately it didn't, and West Ham, in the end, were very pleased that they had won."