The club sacked Allardyce this week, despite him guiding the club away from the relegation zone and into an eighth-place finish in the Premier League.
Everton have been admirers of Silva for some time and the Portuguese was their number one target to replace Ronald Koeman when he was sacked back in October.
However, Watford refused to let their boss go, despite the offer of a significant sum in compensation. Eventually though, Silva was sacked by the 'Hornets' in January, with the club blaming Everton's appraoch for their downturn in form and warding the 'Toffees' off any future approach by threatening legal action.
According to the aformentioned source though, those threats are not an obstacle to appointing the 40-year-old and with initial talks having gone well, he is expected to be appointed early this week.
Silva's appointment could mark the start of another period of upheaval for Everton, who have already pushed director of football Steve Walsh out of the door following their disappointing business last summer.