Manchester United and Chelsea have also been strongly linked with the Portugal international, who has been spectacular since breaking into the Sporting Lisbon first-team.
The versatile midfielder can operate in central midfield or defence, potentially bringing extra cover to an Arsenal squad that has looked threadbare in both positions this season.
But likewise, Manchester United boss Louis van Gaal has money to strengthen in the January market and a defensive signing is believed to be a priority.
The Old Trafford club look efficient going forward but their defence has let them down on several occasions in the first seven Premier League games, famously so against Leicester City.
Carvalho has a £37m release clause in his Sporting contract, and the Portuguese giants have publicly stated that they don’t want to lose their star man, which means this could develop into something of a transfer saga.
The ideal situation for Sporting would be for Manchester United and Arsenal to get involved in a bidding war for Carvalho.
United signed Marcos Rojo from the Liga Sagres club over the summer, and have a long-standing association with Sporting that dates back to the Cristiano Ronaldo transfer. It may give them the edge for Carvalho, but Arsenal reportedly have the march as it stands.