Director of Trading Ian Marmion said: "Marco Silva is still on the lookout for a new club to manage after being sacked by Watford back in January and the 40-year-old is primed for a Premier League return this summer with a number of clubs set to scramble over his managerial services".
The 63-year-old took over the role in December after almost two months of the Blues searching for a replacement for former boss Ronald Koeman.
Everton were unbeaten in their first six league matches under Allardyce and claimed their only Europa League win under the ex-Crystal Palace, Sunderland, Blackburn, West Ham, Bolton and Newcastle boss.
"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", said Everton chief executive Denise Barrett-Baxendale in a statement.
Everton finished the season in eighth position after lying 13th in the table when Allardyce took over but his tenure was characterised by huge criticism from fans.
Silva's Portuguese compatriot Paulo Fonseca, the manager of Shakhtar Donetsk, is second favourite. During his tenure, the club was also forced to apologise to Allardyce last month for a survey that was sent to fans asking them to rank their manager's performance on a scale of 0 to 10.
England's all-time top scorer had reportedly agreed terms on a £12.5 million move to DC United in Major League Soccer last week.
Allardyce was told the news this morning but hit out last night after the club announced sweeping changes to its management structure.
Asked about Everton's current situation, Stubbs told talkSPORT host Jim White: "I'm pretty sure they have someone in mind".