The Sun says new Red Devils boss David Moyes has been a long-term admirer of the midfielder, 22, yet knew he was out of reach of former club Everton.
But the Premier League champions have also tracked Thiago for the past three years.
And they could now prove an irresistible draw for a player who has been frustrated by a lack of first-team action at Barca.
United are willing to offer Barca £15million in a bid to see off rivals Manchester City.
Thiago underlined his potential by bagging a first-half hat-trick in Spain’s 4-2 triumph over Italy to win the European Under-21 Championships last night.