The Hammers were linked with the likes of Carvalho, Renato Sanches and Grzegorz Krychowiak in their search for midfield reinforcements, but in a transfer review on the club's official website Sullivan claimed that they attempted to sign the former before turning down the chance to bring the latter two to the London Stadium.
However, Sporting communications chief Nuno Saraiva has branded the West Ham chairman a liar and claims that no such offer was made for the Portugal international.
"Mr David Sullivan is lying. No proposal arrived for the player William Carvalho," Saraiva wrote on his Facebook page.
"Mr Sullivan has a duty to prove what he says. For this reason, Sporting Clube de Portugal challenge him to publicly show the proposals he says have been made, as well as the evidence that Sporting CP has received them."
The deal was understood to be worth around £27m.