The 21-year-old only joined Sporting Lisbon over the summer, but a string of impressive performances have caught the eye of Europe's biggest clubs.
According to A Bola, Manchester United sent representatives to run the rule over Carvalho during Sporting's game against Gil Vicente at the Estadio Cidade de Barcelos yesterday.
However if the Red Devils wish to firm up their interest in the midfielder then they may have to act quick - the Portuguese newspaper report that Inter Milan, Zenit St. Petersburg, Everton and Anderlecht also had scouts at the game, along with representatives from St Pauli, Celta Vigo, Celtic and Hoffenheim.
Manchester United may also find it difficult to get a deal done, especially in January, because of the complex nature of his economic rights.
Carvalho is only 60% owned by Sporting, with a company named as BES owning the remaining 40%.
The player also has a €45 million release clause, which the Red Devils are highly unlikely to activate.
Manchester United have endured a difficult start to the season under new boss David Moyes, and they currently sit 13 points adrift of Premier League leaders Arsenal following back-to-back defeats at Old Trafford.
However they have the chance to claim tool spot in there Champions League group when they take on Shakhtar Donetsk at home on Tuesday.
Despite the interest from Manchester United and a host of others, Sporting have ruled out letting any player leave - unless a 'ridiculous' offer is made.
"The club has decided that there will be no sales in January," said club president Bruno de Carvalho.
"The club is leading the division for the first time in nine years and we are determined to fight for the title.
"Only if a ridiculous offer is put in for a player will it be studied."