The paper state the defensive midfielder could cost the Red Devils £26 million, a fee which is likely to rule out Tottenham revisiting their own previous interest in the player.
He was playing for Cagliari at the time, before making a permanent switch to Roma just last summer, having joined them on loan six months earlier.
Nainggolan is a Belgian international who has eight caps to his name, competing with Mousa Dembele and Axel Witsel for a place in the side's midfield.
This season he has made 29 appearances for Roma, stepping up in the injury void left by Kevin Strootman, and scoring four goals.
Signing Nainggolan would address the lack of aggression cited by Manchester United boss Louis van Gaal after the club's 3-0 loss away to Everton.
He averages 2.7 tackles per game this season, more than United's most prolific midfielder Daley Blind, on 2.5.