Manchester City and Real Madrid are also thought to be interested in the Uruguayan international. But his Premier League suitors were stunned by the striker's wage demands in meetings with his agents earlier this week.
Cavani is currently with Uruguay at the Confederations Cup, and is due to face Brazil in the semi-final on Wednesday evening.
But Premier League clubs reportedly held talks with his representatives regarding a summer move in the past seven days, only to be shocked at his terms.
The forward recently revealed his displeasure at Napoli's insistence on receiving his £53m release clause, but Cavani himself may have to negotiate if he wants to secure his move away from the Naples club.
His current demands would place a massive financial burden on any club, with a five-year deal at £300,000-a-week worth around £130m.
The same newspaper first reported United's interest in Cavani earlier this week, after it seemed to be a straight race between Chelsea and Real Madrid.
But David Moyes will be supported in the transfer market this summer, and has been linked with several big-name European strikers.
The defending Premier League champions already have Robin Van Persie and Wayne Rooney though, and Cavani's demands would probably place their ability to meet the Financial Fair Play regulations in jeopardy.
City, one of the few clubs in the world capable of affording such extravagant wages, are reportedly out of the race for the South American.
Manuel Pellegrini is thought to have his eye on other targets, although with Carlos Tevez set to depart for Juventus, the new City boss may yet dip into the transfer market for a new forward.