The 63-year-old is set to hold talks with Everton owner Farhad Moshiri.
If true, the latest developments mark the beginning of a series of significant changes at Goodison Park this summer that may include a new director of football in the form of Marcel Brands, the departure of the club's highest earner in Wayne Rooney, and a new chief executive.
Allardyce initially rejected Everton's overture as he wanted a longer deal than the contract until the end of the season that was offered, and indeed The Toffees came back with a two-year deal which he signed.
Sam Allardyce looks like he will be gone this week. Pic Jan Kruger Getty Images
Moshiri will arrive in the United Kingdom on Tuesday for a series of board meetings which will be crucial to determining the way forward for Everton this summer and while the Iranian billionaire admires Allardyce, he recognises the depth of ill-feeling toward him and knows change is required. Generally there's no smoke without fire.
Sam Allardyce's tenure as Everton manager looks to be over amid reports that the club will terminate his contract 12 months early. We'll wait and see after we meet up this week'.
Moshiri has been preparing for life at Everton post-Allardyce since February and the unrest from the fanbase has been noted by the Blues higher-ups.More news: I'm not interested in being the best in history - Messi
Defining moment: The 2-1 victory over Tottenham Hotspur in November which ultimately signaled the end of the Foxes' poor start to the season and saw them start a four-game winning run and leave the relegation battle in the rearview mirror.
Allardyce secured an eighth-place finish for Everton in the Premier League after just two defeats in their final nine league matches.
Silva was sacked in December by the Hornets and in a statement released by the club, there was a swipe at the Blues.More news: Rafael Nadal vs Fabio Fognini: Italian Open quarterfinal preview and prediction
Speaking about his future on Monday, he told the publication: "It's my job to turn that around".More news: The new Google News app is now widely available