A statement from Everton made no mention of Silva, instead saying the search for a new manager would commence immediately as well as thanking Allardyce for his work.
The club apologised to Allardyce in April after a survey was sent to fans asking them to rank their manager's performance on a scale of zero to 10.
Marco Silva is thought to be the frontrunner for the Everton post once more, despite their approach for the then-Watford boss past year causing acrimony between the two clubs and preceding the Portuguese coach's departure from Vicarage Road.
Allardyce, who was briefly England manager in 2016 before returning to club management at Crystal Palace last season, had already reacted angrily to the structural changes at the club prior to his meeting with majority owner Iranian-born British tycoon Farhad Moshiri.
'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 Metro.
The Toffees announced a major restructure of their board on Tuesday, as Barrett-Baxendale replaced Robert Elstone as chief executive.
Asked about Everton's current situation, Stubbs told talkSPORT host Jim White: "I'm pretty sure they have someone in mind".
"I came into the football club with the team struggling and we have finished 8th in the table".
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 would 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", the statement said.