The Gunners were interested in the 22-year-old midfielder all summer, and according to Portuguese news outlet A Bola, failed in the late bid.
In the offer, Arsenal included the Costa Rica striker Campbell, who has failed to settle at the club since returning from his loan spell at Olympiakos this summer. Campbell had also been linked with a move to fellow Portuguese side Benfica.
A Bola have reported that the Gunners put in the late player-plus-cash bid for Carvalho on Monday, but were told that the Portugal international was not for sale.
Carvalho himself told A Bola on deadline day: "I am under contract with Sporting and I'm a Sporting player.
"There is no way I can say I am going to leave. I am fine at Sporting."
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies