The former England manager signed a contract until June, 2019, when he took over in November after Koeman recorded strings of poor result.
The Toffees went on to finish 8th in the league but the former England boss never really received widespread approval from Everton fans.
Speaking to the Mirror at the League Managers Association annual awards, Allardyce made his anger at not being informed of those decisions known and suggested the writing was on the wall regarding his own future.
Everton scored only 44 goals in 38 outings in the league, the second lowest tally of all teams outside the bottom six, and with the future in mind, the club have decided to make a change.
"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", he told the Liverpool Post on Tuesday.
Professor Denise Barrett-Baxendale was promoted to chief executive and Allardyce said he was "disgusted" not to have been informed.
'Sam was brought in at a challenging time last season to provide us with some stability and we are grateful to him for doing that.
Theo Walcott was a player that many supporters of Everton where happy to see and that deal cost some £20m, so he has a big task ahead of him to prove he was worth that amount of money, but he is of course a top-level player and he should take the move in his stride.
"Again, we'd like to place on record our honest thanks to Sam for his work with us over the last few months and wish him well for the future".
Everton have also been interested in Shakhtar Donetsk coach Paulo Fonseca, although he is also a contender to replace David Moyes at West Ham.