The Red Devils are leading the race to sign their long-term target after years of speculation.
Napoli could be forced to sell their vice-captain if they fail to reach the Europa League due to concerns about their funds.
United signed Harry Maguire in a world-record deal for a defender last summer after an £80million move to Old Trafford from Leicester City.
The England international was appointed as club captain after Ashley Young joined Inter Milan last month.
Maguire has failed to inspire United to put together a top-four challenge as the club struggled at the back.
Koulibaly has a whooping £127m release clause on his contract at the Naples outfit.