The Sun says Moyes sees the Spanish international, 22, as a possible replacement for the retired Paul Scholes.
Thiago has two years to go on his £30,000-a-week Barca deal but is keen to quit after falling out of favour.
A Catalan insider said: “He’s not happy because of his lack of playing time and wants to go to a club where he’ll get regular first-team football.
“Boss Tito Vilanova is not a big fan and has marginalised Thiago this season, preferring to play Cesc Fabregas or Alex Song ahead of him.
“That frustrates Thiago because he knows he is good enough to probably get into any other team in the world.”