Wayne Rooney's future at the club was reported to have rested on whether Allardyce would be manager next season.
"We have made the decision that, as part of our longer-term plan, we will be appointing a new manager this summer", said new chief executive Denise Barrett-Baxendale.
"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. The 40-year-old Portuguese was the preferred choice for owner Farhad Moshiri when Ronald Koeman was shown the exit door. He was subsequently sacked from that role as results slipped, with the Watford board claiming that Everton's overtures had caused the slump.
The Goodison Park club are eager to appoint Silva as the successor to Allardyce, but are now involved in a dispute with Watford over an approach made for the Portuguese while he was still at Vicarage Road.
"I can't quite say confident after all the rumours I'm reading in the paper", said Allardyce, when questioned about his future.
Whoever is appointed, Stubbs says they face a huge summer as they look to amend the mistakes Everton made in previous transfer windows, having had little to show for the nearly £300million spent since Farhad Moshiri's takeover.
Also put out on loan were Sandro Ramirez who is now settled at Sevilla and Ademola Lookman who is earning his keep over at RB Leipzig, another player that has been loaded out is Mo Besic who I think is a very welcome addition to the Middlesbrough team.
The Everton role was Allardyce's 11th permanent managerial post but fifth since 2015.