Sky Sports News HQ reported "United's interest is at an early stage and no deal has been agreed."
Chelsea are said to have had several bids for the Belgian rejected last year, the largest of which is believed to have been worth more than £34 million.
United could do with recruiting another central midfielder who can strengthen their spine alongside Paul Pogba and Ander Herrera.
Football writer Liam Canning highlighted their need to bolster their squad before they can compete with Europe's elite once more:
Nainggolan has had yet another excellent year with Roma, contributing 14 goals and six assists in all competitions.
According to Squawka, the all-action midfield general has created 64 chances in Serie A this year, as well as winning 55 tackles and 37 interceptions.
A dynamic and combative presence in midfield, Nainggolan is a robust ball-winner who can effectively support attacks with his outstanding creativity and eye for goal.
GianlucaDiMarzio.com's David Amoyal believes he hasn't quite been at his best this year, but beIN Sports' Matteo Bonetti is a huge admirer:
Nainggolan only turned 29 earlier in May, so while he wouldn't be a long-term solution, he could still provide at least two or three years of top service at Old Trafford.
He could be pricey, but his signing would be another strong step in the right direction for the Red Devils.