The Bosnian playmaker has been a long-term target for Tottenham, and was linked with a summer move to White Hart Lane.
However Spurs were unable to tie up a deal, leaving the door ajar for United.
David Moyes has been impressed by the midfield schemer this term, with his performances helping Roma to top the Serie A table as it stands with nine wins from nine games.
The Scot is desperate to add some creativity to his ranks, and sees the 23-year-old as the perfect man to solve his problems.
Pjanic is set to enter the final 18 months of his contract this winter, which could tempt the Italian giants to sell rather than allow his value to drop.
An extension has been mooted, but it remains unclear as to whether the player – who has had his problems at the Stadio Olimpico – is looking to stay.
Pjanic has gained a reputation as a set-piece expert during his career, thanks in part to the influence of former Lyon team-mate Juninho Pernambucano – who is widely regarded as one of the best dead ball players on the planet.
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