Allardyce was attending the annual League Managers' Association awards when news broke that the Toffees had brought in a new CEO and deputy chairman.
The 63-year-old said that he was "disgusted" by the behaviour of the board after they failed to inform him of the imminent changes, and suggested that his own future is likely to be decided during a meeting with club owner Farhad Moshiri today.
"I'm shocked, disappointed and disgusted that the football club didn't have the decency to tell me, my director of football and my staff about the changes," Allardyce is quoted as saying by The Mirror.
"They must have been in the pipeline for a considerable time but no-one thought to tell me and my staff. I'm pretty disgusted. I came into the football club with the team struggling and we have finished eighth in the table. I'm more than happy with what myself, my staff and the players have achieved from when I came in.
"I will be meeting Farhad on Wednesday morning to discuss what happens next but I'm not expecting a long meeting."
Marco Silva is currently the favourite to take over from Allardyce as Everton boss.