The 23-year-old has been on the Red Devils’ radar for some time, but they missed out on his signature over the summer as he switched to the Serie A giants from PSV Eindhoven.
Since the move Strootman has been impressive, helping the Stadio Olimpico-based side into the upper echelons of the Italian top flight.
His impressive form has alerted David Moyes, who is ready to launch an audacious move to persuade the 23-year-old to cut his stay in Rome short next month.
The Scot is desperate to land a combative midfield man to bolster his engine room after watching his new side make a woeful start to the season.
Moyes is thought to see some of his current central options as not up to scratch, and feels that Strootman could add a feisty edge to his side.
Summer arrival Marouane Fellaini was initially believed to be the player designed for the role, but the Belgian appears to be more suited to an attacking position, and has struggled since his switch to Old Trafford.