Sky Sports says United boss Jose Mourinho is keen to bring the talented midfielder to Old Trafford, although a concrete offer is yet to be tabled.
Mourinho's former club Chelsea were reported to have had three bids rejected for the 29-year-old playmaker during last summer, with the final offer worth £33m ($43m).
Nainggolan has scored 11 goals in 35 Serie A appearances for Roma this season. He has more than three years left to run on his contract at the Stadio Olimpico.
The Belgium international confirmed in March that he spoke with Chelsea boss Antonio Conte last summer about a potential transfer.
"Yes. I talked to him," Nainggolan told Sky Sports. "They were really interested and wanted me over there. That's all I can say."
The 29-year-old revealed that he was a fan of the Premier League and refused to rule out a move to England in the summer.
"I watch [the Premier League] a lot. A lot of my teammates play there. It's the best league in the world," he said.
"We have to make choices in life. I am having a good season. When the window opens we will talk about it again."
Mourinho has suggested that he will bring in several new players in the summer transfer window to reinvigorate a United squad that is headed to an underwhelming sixth-place finish in the league.
Atletico Madrid forward Antoine Griezmann, Torino forward Andrea Belotti, Burnley defender Michael Keane and Tottenham midfielder Eric Dier are among the players linked a summer switch to Old Trafford.
Mourinho vowed that the Red Devils would challenge for the title in the 2017/18 season on 19 May.
"If, in the next transfer window, we manage to be successful, I think we can [challenge for the title]," he was quoted as saying by Sky Sports.
"There is work to be done. The good thing is the club knows, the board knows, I know and we are together on this process."