The Toffees dismissed Ronald Koeman just under a month ago the chances of caretaker boss David Unsworth remaining in charge until the end of the season continue to increase.
According to 'Sky Sports', former England boss Sam Allardyce ruled himself out of the running after talks with the Merseyside club.
Allardyce felt that Everton's contract offer, which would only run until the end of the season, wasn't enough to tempt him out of retirement.
Everton also contacted Watford for a second time on Monday morning, and offered to pay compensation for Silva's services, but the Hertfordshire club would not let their Portuguese coach speak to the Toffees.
Reports suggest that Silva would be interested to talk to Everton, but his two-year contract at Vicarage Road does not include a break clause, so the Goodison Park club will have to look elsewhere.
A spokesman for Watford told 'Sky Sports': "We will not countenance an approach for our manager, either from Everton or indeed any other club, be it in the Premier League or abroad."
It is also understood there has not been any contact between Everton and Burnley over Sean Dyche, who was the early favourite to replace Koeman.