Silva was named Toffees manager last week following the departure of Sam Allardyce at the end of last season and has since identified the Magpies centre-back has his first potential signing.
The former Watford manager wants to take advantage of the uncertainty surrounding the future of Newcastle manager Rafa Benitez and the continued ownership of the club by Mike Ashley.
Benitez, a fan-favourite at St James Park, has yet to agree a new contract at the club after comfortably steering them to Premier League safety last season with the Spaniard harbouring concerns over his backing in the transfer market.
According to The Sun, Everton will test Newcastles resolve by making a bid of around £30 million for the 24-year-old defender.
Silva wants Lascelles to be the long-term central defensive partner for Michael Keane, who was signed in a similar deal with Burnley last season but has yet failed to live up his £30m price tag on Merseyside.
The Portuguese is believed to be concerned at the performances of ageing Everton centre-backs Phil Jagielka and Ashley Williams.
Chelsea have also recently been linked with a move for Lascelles, but those reports have cooled off as the Blues continue to delay the inevitable departure of Antonio Conte and the appointment of a new manager.