David Sullivan said the London side had a bid for the 25-year-old Carvalho rejected by Sporting, before accepting a deal, and then being turned down by the joint-chairman of the Hammers.
But communications chief Nuno Saraiva says those statements are untrue.
Sullivan said on the West Ham website on Friday: “It is no secret that we made a club-record bid for Sporting Lisbon’s William Carvalho but unfortunately our offer was rejected a couple of weeks ago.
“Late last night, Sporting Lisbon made contact to accept the original offer but, unfortunately, it was just too late in the day, and we simply did not have enough time to put the player through a medical.
“We were not prepared, as a club, to buy a player for that amount of money without him having gone through adequate medical checks.”
Saraiva responded with his comments on Facebook: “No proposal arrived for the player William Carvalho.
“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.”