The new Old Trafford manager has instructed chief scout Jim Lawlor to get the wheels in motion regarding a summer transfer, and the report claims United have already made contact with the player's representatives.
Thiago's father - former-Brazil international Mazinho - is thought to be willing to hear United out, and the Premier League champions hope to be able to put together a bid suitable for Barcelona and the player in the near future.
Thiago is currently with the Spanish under-21 side at the European Championship in Israel.
And the report claims he's ready to end his time at the Nou Camp having grown frustrated at a lack of playing time.
The 22-year-old made his first-team debut in 2009 but has only made 68 first-team La Liga appearances since then - the majority has a substitute.
He's a full Spain international having won three caps for his country, but United could face a fight for his signature with title rivals Manchester City.
The two Manchester clubs are competing for the young midfielder, but Moyes hopes the promise of regular first-team football will stand in his favour.
The former-Everton manager has been a long-term admirer of Thiago dating back to his time at Goodison Park.
But, now at United, Moyes hopes the club's greater financial firepower will get the ball rolling and he's taken the first step to steal a march on City.