The Daily Mail report that Everton officials contacted Watford on Monday morning, having made the 40-year-old former Hull City boss their number one choice to take up the reins at Goodison Park.
He nearly saved Hull from relegation from the Premier League and since taking charge at Watford he has guided them to their best ever top-flight start since the Graham Taylor era.
It is understood Silva would like the opportunity to talk to the Merseyside club and chairman Bill Kenwright and owner Farhad Moshiri, but the lack of a release clause in Silva's contract means he is bound to the club for the two years he signed on for during the summer.
That Everton have chose to make any kind of formal move would suggest a degree of confidence Silva is interested in talking to the Goodison board.
The Portuguese has been highly praised since arriving in England 11 months ago.
The arrival of Silva at Watford, replacing Walter Mazzarri, sparked an instant improvement in their form but they have lost their last three Premier League games and have slipped down to ninth in the Premier League table.
Koeman has been temporarily replaced by caretaker boss David Unsworth, and whilst he got his first win with a 3-2 comeback victory over Watford before the global break, Everton have been looking elsewhere too.