Lukaku, 23, scored 25 goals for Everton this season and has attracted the interest of a number of sides.
He joined the Toffees from Chelsea after a season-long loan stint that came off the back of a successful loan at West Brom in the 2012/13 season.
Chelsea are thought to be interested in re-signing the Belgium international, while Juve consider him a like-for-like replacement for Alvaro Morata, who could leave for Real Madrid.
Paris Saint-Germain have also kept track of his recent performances.
Everton reportedly want £65million for Lukaku, who cost them £28m when he switched from Chelsea on a permanent deal in 2014.
Incoming Stamford Bridge manager Antonio Conte - currently head coach of Italy - is said to be a fan of the striker.
But new Everton owner Farhad Moshiri is thought to have set aside £100m for new transfers and the club are under no obligation to sell Lukaku.
Express Sport understands Everton are in talks with Southampton over a £4m compensation package to appoint Ronald Koeman as their new manager.
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As for Juventus, they are on the verge of completing a three-year deal to sign Dani Alves from Barcelona.
That is according to Italian newspaper Tuttosport, who suggest Lukaku could swiftly follow Alves to the Serie A champions.