The Uruguayan international is believed to prefer the idea of moving to the Premeir League champions should he decide the English top flight is the best place to play his football next season.
Chelsea have been long-term suitors, and the report claims Jose Mourinho will find out if they still have a chance of signing the forward in the next few days.
Cavani is expected to hold talks with Napoli president Aurelio De Laurentiis this week after returning from the Confederations Cup following elimination at the semi-final stage.
But if offered the opportunity to play in England, the report claims Cavani would favour a move to United.
Their interest in the player has only surfaced recently, but they pose a legitimate challenge to Chelsea, as well as Manchester City and Real Madrid.
City could still re-enter the race after selling Carlos Tevez to Juventus for £10m, but reports today claim they're more likely to rekindle interest in Borussia Dortmund star Robert Lewandowski.
Madrid have also been linked, and kicked off their summer transfer activity by announcing the capture of Malaga star Isco early on Thursday - the Spain under-21 playmaker was also a target for City.
De Laurentiis is refusing to entertain the notion of selling Cavani for less than his £53m release clause - but the report claims none of the interested clubs are willing to go that high.
The player has hinted at leaving Napoli at various points this summer, and even had to ask his parents to stop commenting on his future after they dropped him in trouble with his current club.