Despite touting retirement or another chance at global football management, Allardyce could be easily lured back to the Premier League with the exciting prospect of turning around the fortunes at Everton. If Marco Silva was coming in, they might feel a tad more excited.
Allardyce was an early candidate to succeed Ronald Koeman, who was sacked on October 23, but publicly withdrew after Everton delayed making an offer.
Five weeks since sacking Ronald Koeman, Everton have still not appointed his long term replacement.More news: Uber-Waymo case: Dramatic turn as judge says Uber lawyers withheld evidence
The Toffees are desperate to get a new boss in place after the 4-1 defeat to Southampton in the Premier League yesterday.
Big Sam had resisted Everton's initial interest - first revealed by SunSport last month - when the Toffees were only willing to give him a job for the rest of the season before trying for a younger, sexier name.
Sources close to the Portuguese have told Football Insider that Fonseca was offered the Everton job but has rejected it to stay in Ukraine with Shakhtar.
Interim manager Gary Megson will oversee proceedings against Newcastle on Tuesday night - with Pardew expected to be in the stands ahead of an announcement after the game.More news: MacOS gives users full admin rights without password
"I think there has been a generation of really good players out there who I believe should go and manage and try and come up".
Although Allardyce's qualities as a coach versed in ensuring the Premier League survival of a series of troubled clubs has been pushed by sections of Everton's dysfunctional hierarchy, those behind the club's recent takeover have tried to avoid appointing an individual whose style of football and management does not mesh with their own vision for the club.
West Ham are ready to make a move for Krasnodar striker Fyodor Smolov, and, at the same time, according to the Liverpool Echo, Everton have been linked with a move for Besiktas playmaker Oguzhan Ozyakup, who was once dubbed "the new Zinedine Zidane".
West Ham will kick-off in the relegation zone but a victory would see them climb above the Toffees.More news: Instant Peay Play: APSU Volleyball heads to UCLA for NCAA Tournament opener
"When confidence is low, it takes a bit of time to build that back up".