Jose Mourinho is thought to be an admirer of the Italian international, and the 'Red Devils' would be prepared to double the player's salary which currently stands at close to £890,000 per year.
According to 'Corriere dello Sport', Pellegrini's agent Giacomo Pocetta met Manchester United directors in London on Saturday to discuss a possible deal for the player, whose contract runs out in 2022.
Pellegrini, 22, is, however, seen as a key player at Roma. The Serie A side's director of football, Monchi, will reportedly offer the midfielder a new contract in an effort to remove or increase the release clause and ward off other clubs.
Roma already lost Radja Nainggolan and Kevin Strootman in the summer, to Inter Milan and Marseille respectively, and will be reluctant to let another of their midfield stars leave the club.
Pellegrini has scored twice and assisted five more for Roma this season, and is also becoming a regular at international level.