According to RMC Sport (via the Mirror), the Red Devils are not prepared to become embroiled in long negotiations for the Algeria international left-back as his contract runs to 2022, but they have already been in contact with Ghoulam's agent, putting them a step ahead of rival Premier League suitors.
The 27-year-old has been out of action since the beginning of November due to a knee injury, but he is set to return to training this week, the report added.
He has been a stalwart in Maurizio Sarri's exciting Partenopei side since the Italian took over as manager in 2015.
United are short on options at left-back. Shaw looked set to become a regular at Old Trafford when he was signed from Southampton as an 18-year-old for £27 million in 2014.
However, he has become a peripheral figure under manager Jose Mourinho, and after recent criticism from the Portuguese boss he does not look set to remain at Old Trafford for much longer, per Simon Stone of BBC Sport.
Daley Blind and Matteo Darmian are also unreliable and could leave this summer, while Ashley Young, who has impressed on the left side of defence in 2017-18, turns 33 in the summer.
Ghoulam could be a fine option in the United side to take up the starting left-back role in the long term.
He has a wealth of experience both domestically and in Europe, and he boasts the attacking and defensive skill sets required of any top-class full-back in the modern game.
The issue for United is whether they can persuade Napoli to let Ghoulam go.
He is a key player for the Serie A outfit when fit, and it is likely the Red Devils will need to offer a lot more than £26 million in order to get their man.