United boss David Moyes worked hard over the summer to sign a top-tier midfielder, with failed bids for Cesc Fabregas, Ander Herrera and Sami Khedira.
Although the Old Trafford club signed Marouane Fellaini on deadline day, the report claims Moyes is keen to add to his squad again at the earliest opportunity in January.
And they will look to beat Tottenham to the capture of €30m-rated Pjanic. The Bosnia-Herzegovina playmaker is out of contract at Roma in the summer of 2015, but Italian paper Gazzetta dello Sport suggests he will be offered an extension.
Still, if he rejects or stalls on the new deal, Roma will be forced to cash in on Pjanic. Next summer he'll have less than 12 months to run on his contract and the Italian club would be forced to accept a reduced fee.
Pjanic has been impressive for the current Serie A leaders this term, with three goals and three assists from nine appearances for Rudi Garcia's side.
United, in contrast, have been struggling - although they did record an important come-from-behind home victory over Stoke City at the weekend.
Despite the addition of Fellaini, Moyes feels the squad needs strengthening in the centre of the pitch.
Pjanic is thought to be the top target, but United could face competition from Spurs boss Andre Villas-Boas. The Tottenham manager was linked with Pjanic over the summer and a recent talkSPORT report claims Spurs are still interested heading into the January window.