United have been tracking the Bosnian playmaker for some time, and Moyes had originally planned a January bid until Juan Mata became available.
But with the Premier League champions seemingly no closer to finding their form, it promises to be a summer of rebuilding at Old Trafford, and Pjanic has remained a target.
Moyes is keen to start a major overhaul of his squad, and the Glazer family are ready to back him heavily in the transfer market in order to avoid a repeat of this season.
Roma attacker Pjanic is under contract until 2015, and has previously been linked with Barcelona.
The Serie A side is desperate to tie him down to a new contract, and are set to offer improved terms.
Talks have been pencilled in this week between the player, his agent and Roma sporting director Walter Sabitini.
It remains unclear as to whether the 23-year-old wants to stay at Roma, but the club are hoping a bumper pay rise will be enough to see off the interest of Manchester United and Barca.