The Nigerian is currently on a season-long loan at Galatasaray where he has netted 28 goals in all competitions alongside five assists.
Osimhen has no plans to continue in Turkey next season and he would prefer a high-profile transfer to the Premier League instead.
ESPN claim that Chelsea and United are interested in the Nigerian star, who could be signed for around £56m from Napoli in the next transfer window.
The Serie A outfit won't sell him for anything less. Osimhen is prepared to move to England for an annual salary of £9.4m after tax.
United have him on their shortlist for the summer, but it is claimed that they could prioritise a deal for Liam Delap from Ipswich Town.
The 22-year-old will be available for £40m once Ipswich are relegated. He could be signed for a relatively affordable salary package.