It now seems that the Toffees are set to appoint a new manager soon with Unsworth confirming that there are talks taking place between the club and a few candidates.
The pair worked under the veteran Premier League manager during his most recent spell in management with Crystal Palace which ended in May.
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Conte also praised Swansea manager, Paul Clement who was an assistant to Carlo Ancelotti at Chelsea during the 2010/11 season. "He is in contention but at the same time I have to make the best decision for every player".
When he left Selhurst Park in May, Allardyce said he had "no ambitions" to take another managerial job.
It is also understood that Shakhtar Donetsk manager Paulo Fonseca was also interviewed for the role, while Republic of Ireland's Martin O'Neill was also being considered, but Everton owner Farhad Moshiri was expected to make a firm decision on Tuesday.
A loss would see Everton sucked into the relegation zone.
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Everton have lost seven of their 14 league games this season and can not qualify for the knockout stages of the Europa League after earning one point from their five Group E matches.
Despite claiming a 3-2 win over Watford, the Merseyside club suffered defeats to Leicester City (2-0) and Southampton (4-1) in the league while also crashing out of the Europa League thanks to humiliating losses at the hands of Lyon (3-0) and Atalanta (5-1).
His first game will be at home to Huddersfield on Saturday, and he will no doubt want to cap his first game off with a victory.
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The Toffees hierarchy will feel they have gone for a safe pair of hands, with Allardyce having taken over both Palace and Sunderland in graver situations and led them to safety. But that turned sour after only one match, with Allardyce stepping down in September 2016 following a newspaper sting.