Big Sam is a favourite amongst bookies to take over, with Everton maybe looking to give to former England manager a chance to manage the Merseyside club.
They were hoping to secure him on a deal until the end of the season, with sources claiming they planned to return for Silva next summer, but Allardyce is keen for a longer-term contract to offer more job security.
Unsworth has overseen five defeats in seven games since becoming Ronald Koeman's temporary successor on 23 October, with the latest, Sunday's 4-1 defeat at Southampton, increasing Everton's desperation to make a permanent appointment.
With the club now just two points above the bottom three ahead of Wednesday's encounter with third-bottom West Ham at Goodison, there is a determination at Everton to resolve the managerial uncertainty in the coming days.
David Unsworth - the club's under-23s manager, who had two spells at Goodison Park as a defender - has been in caretaker charge.
A move to Watford, advised by fellow countryman Jose Mourinho, has seen Silva pick up impressive results which have caused Everton to make multiple inquiries about his availability regarding a switch from being the Hornet's boss for just four months.
Unsworth added: "I think we need our best players fit, quickly, some proven Premier League recruitment in early January and I think we need to win a couple of games and get confidence flying again".
And Everton failed to replace Romelu Lukaku who made the switch to Manchester United.
The search for a replacement boss then descended into farce as Allardyce initially pulled out of talks because he wanted a deal beyond the end of this season, which major shareholder Farhad Moshiri and chairman Bill Kenwright did not want to offer. It will be "you do what I tell you".
"If he doesn't take the Everton job I think I would tell "Unsy" to go and manage somewhere and earn your corn", said Moyes.
West Ham will kick-off in the relegation zone but a victory would see them climb above the Toffees.
Having spent such a short amount of time managing in the Premier League, Portuguese man Marco Silva has rapidly built up a strong reputation as one of the league's most talented young managers.