The central defender is Jose Mourinho's top defensive transfer target for the January window.
Senior Old Trafford officials recently met with the Serie A club to talk about the chances of a mid-season transfer, say the Sunday Times.
Even despite the price tag, United could face difficulties in the form of Napoli owner Aurelio De Laurentiis and manager Carlo Ancelotti.
De Laurentiis' club, second in Serie A, are under no great financial pressure to sell.
Ancelotti took the gig in the summer on the understanding he would hold a veto over the sale of any senior player.
It is further claimed that 27-year-old Koulibaly is encouraged by Mourinho's interest.
However, the player is contracted to Napoli until 2023 and Mourinho reportedly failed recently with a £77m bid for him.
He agreed a new deal at the end of the summer window and speaking of the contract, De Laurentiis said: "I erected a wall to keep him here with me."
United looked solid in a 4-1 thumping of Fulham yesterday but Mourinho has said for a long time that his defence needs strengthening.
He joked there were no water bottles in his post-match press conference because the club were saving money for a splurge in the January sales.
United have conceded 26 goals in 16 Premier League fixtures so far this season but Mourinho is uncertain if executive vice-chairman Ed Woodward will sanction spending so much in January.
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