The Toffees made a second bid of £8.5million to try and tempt Silva away from Vicarage Road, it has been reported.
It is suggested that the Hornets are braced for a third approach in the coming days but they are desperate to keep hold of their manager.
Everton spoke to ex-Bolton and England manager Sam Allardyce at the weekend but negotiations are thought to have hit at a crossroads.
Silva is much sought-after having guided Watford up to ninth in the Premier League table.
And the Daily Mail report that he wants to speak to Everton chairman Bill Kenwright, but no release clause in his two-year deal, which was signed in the summer, is proving a stumbling block.
Silva has been highly successful since turning his hand to football management.
The former right-back won the Taca de Portugal when starting out as boss of Sporting Lisbon.
He then moved to Greece outfit Olympiacos where he won the league title in his first season with a run of 17 straight victories.
A move to Hull City in the Premier League is the only blemish on his record, but he did reasonably well when trying to steady a sinking ship having joined in January, but ultimately they suffered relegation.
Crystal Palace did register an interest in Silva in the summer, but they went for Frank de Boer instead.
If Silva was to join Everton, he would take over a side six places and four points behind his current team.