Newcastle United and Bayern Munich are said to be interested in the 23-year-old Nigerian international who won the Serie A Best Young Player award last season.
Osimhen has scored 24 goals in 51 Italian top-flight games for Napoli and is valued at £85m, according to reports.
He has 22 caps for Nigeria and has netted 15 times for the Super Eagles.
Osimhen’s glowing form at Napoli has seen him linked to some of the biggest clubs in Europe, but he played down links with a move away from the Stadio Diego Armando Maradona this summer.
‘I’m in Napoli. And I have great respect for my club,’ Osimhen told Corriere Dello Sport.
‘They are just rumors of the market. I’m fine here and I’ve never had such close relationships with everyone as at this moment.
‘I spoke to the president, he is the one who decides, and he reassured me by explaining the club’s plans.
‘I am happy with what he told me and the purchases are of quality: words were followed by deeds. I am very happy to play with Napoli and we will see in the future.’
It has been a busy summer for United and Arsenal as both sides look to improve on last season and return to the Premier League top four.
United has already secured deals for Christian Eriksen, Tyrell Malacia, and Lisandro Martinez.
Arsenal, meanwhile, has brought in striker Gabriel Jesus and defender Oleksandr Zinchenko from Manchester City for £45m and £32m each.
The Gunners have also signed Fabio Vieira from Porto for £34m as well as goalkeeper Matt Turner from New England Revolution and Marquinhos from Sao Paulo for undisclosed fees.
United host Brighton in their first Premier League game of the new season on August 7 while Arsenal travels to Crystal Palace on August 5.