United and Arsenal have both sent scouts to Portugal in recent weeks to keep tabs on the highly-rated 21-year-old.
And despite reports of a £40m release clause in his contract, Sporting Lisbon will let him go for just £12.5m as they struggle with financial troubles.
The Portuguese club have a close relationship with United having previously sold promising academy stars to Old Trafford. Cristiano Ronaldo and Nani moved to the English giants over the last decade and flourished.
But Arsenal are also keen on a player who has been compared to one of their modern-day legends, French midfielder Patrick Vieira.
The fact Carvalho is on just €2,000-a-week at Sporting - one of the club's lowest earners - could open the door to an improved offer from the Premier League duo.
The young midfielder has been a key figure in Sporting's impressive start to the league campaign this term, and he was rewarded with his first international cap in the second-leg of the play-off against Sweden last month.
United boss David Moyes is expected to strengthen his midfield in January - despite the addition of Marouane Fellaini over the summer.
Likewise, Wenger may look to supplement his options in the winter market - although a striker is believed to be a priority.