The Red Devils are currently short of attacking options, with Edinson Cavani, Jesse Lingard and Juan Mata all leaving the club on free transfers following the expiration of their contracts.
Cristiano Ronaldo, meanwhile, has allegedly told the club that he wants to leave this summer, and the Portugal international's exit would further deplete the options in the final third for new head coach Erik ten Hag.
Osimhen's future has also been the subject of speculation this summer, with the 23-year-old regarded as one of the best young talents in world football due to his recent goalscoring exploits.
The Nigeria international found the back of the net on 18 occasions in 38 games for Lille, in addition to registering six assists, before arriving at Napoli in September 2020.
Osimhen has netted 28 goals and registered nine assists in 62 outings for his Italian side, and a number of clubs have been linked with his services this summer.
Arsenal are said to be admirers of the Nigerian, while a recent report claimed that Bayern Munich viewed him as a possible replacement for Robert Lewandowski, who is being targeted by Barcelona.
However, according to Gazzetta dello Sport, there is also genuine interest from Man United, with Ten Hag's side keen on his signature amid the Ronaldo speculation.
Osimhen still has three years left to run on his contract in Naples, and there have been suggestions that his Serie A club value him in the region of £80m, which is an unrealistic fee for the Red Devils this summer.
However, Ronaldo's exit would open up a lot of space in the wage budget, and the club could move for Osimhen if they are unsuccessful in their attempts to sign Antony from Ajax.
Last month, Osimhen opened up on the speculation surrounding his future, telling Calciomercato: "I know there are many people that would like me in Spain or England, but I don't think it's the right time to talk about it now that I'm with the national team.
"I always go out on the pitch to do my best both with the national team and with my club.
"I don't know my future, I know anything can happen, but now I just want to go on holiday with my family, clear my head and recharge my batteries. Then we'll take care of everything else."
Barcelona and Real Madrid are also believed to be admirers of the forward, but neither club are believed to be preparing to move for Osimhen during this summer's transfer window.
The striker also boasts an impressive scoring record at international level, finding the back of the net on 15 occasions in 23 appearances for the Super Eagles in all competitions.