The Saudi Pro League side have also offered the Nigeria international a staggering £1m-a-week in wages.
Al Hilal recently saw a bid of €300m (£259m) for Paris St-Germain's Mbappe accepted, only for the player to turn the move down.
Sky Sports reports they have now turned their attentions elsewhere, and are hopeful of securing the services of Osimhen.
The 24-year-old has been a long-term target for both Manchester United and Chelsea.
Erik ten Hag's preferred choice, though, is Atalanta's Rasmus Hoijland, who looks to be heading to Old Trafford in a £72m deal.
Osimhen has also been linked with the likes of PSG and Bayern Munich.
He is yet to extend his deal with the Serie A champions, with the club failing to meet the players demands.
Napoli president Aurelio De Laurentiis has revealed the club are prepared to cash in should he refuse to sign a new contact.
"We have a basic agreement on a two-year extension, but [we are] willing to consider an offer we could not refuse," he said.
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Osimhen finished as the top scorer in Italy last campaign, with 26 goals in 32 games and led Napoli to their first Scudetto in 33 years.