The Dutch midfielder has been on the Red Devils’ radar for some time, with speculation linking them with a swoop for his signature having been populating gossip columns over the course of the past few years.
However their interest cooled over the summer, allowing Roma to snap him up from PSV Eindhoven.
But with Marouane Fellaini failing to settle at Old Trafford, Moyes is ready to re-visit the tough-tackling star.
Strootman has been lauded for his combative playing style, which has drawn comparisons with former United icon Roy Keane.
Having only signed the 23-year-old last year, Roma are far from keen to see him walk away.
Yet they are believed to be willing to do business for around £33m – over double the £15m fee they paid for him a matter of months ago.
Moyes has a large transfer budget at his disposal for the summer, and might meet the Serie A giants’ valuation in a bid to rectify the issues in his midfield.
Strootman could be a long-term successor to Michael Carrick and Darren Fletcher, both of who are edging towards the ends of their careers.