The Hammers were in pursuit of the Sporting Lisbon midfielder throughout August.
But they struggled to reach an agreement over a fee.
West Ham were prepared to make a club-record move for Slaven Bilic’s top target, but the Portuguese club continued to raise their demands.
Fans of the east London club were left frustrated after nobody was signed at the London Stadium on deadline day.
Sullivan then claimed loan deals for the likes of Renato Sanches and Grzegorz Krychowiak were snubbed by Bilic.
Bizarrely though, the Hammers chief insisted Sporting did accept a bid for Carvalho but it was too late to do a medical.
He told West Ham’s official website: "Overall Slaven is happy with the business we have done during the summer transfer window.
"It is no secret that we made a club record bid for Sporting Lisbon’s William Carvalho but unfortunately that 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.
"Grzegorz Krychowiak and Renato Sanches were both offered to the manager before their switches elsewhere, but he told us that he is happy with the squad he has.
"As a Board we are behind Slaven, and he believes he has the tools to turn around our form and rectify our disappointing start to the season."
Bilic is under pressure after a poor start to the new season, but it is Sullivan who is under fire now.
In response to his statement, Sporting’s director of communications Nuno Saraiva has said: “David Sullivan lies.
“At Sporting, as has already been said by our president, no proposal has been made for the player William Carvalho.
“The football industry is not the set of an adult film in which all obscenities are allowed.
“So the boss of a club demands a lot more than this intellectual pornography.
“Mr David Sullivan has a duty to prove what he says. That is why Sporting challenges him to publicly show the proposals he claims to have made as well as the evidence that Sporting received them.
“One thing is certain: Sporting, has already been said, does not cut legs to its athletes. But we have nothing to do with the performance of entrepreneurs or relatives who live at the expense of players and who are offering them a club at a club price, without a mandate, as if they were cheap supermarket merchandise.
“Athlete and the club, and creating regrettable situations that have already forced the Sporting to take drastic measures to defend these characters who parasitise the lives of some players.”
With West Ham fans angry at matters on and off the pitch, a hostile atmosphere is anticipated for the first game at the London Stadium since May when Huddersfield visit next Monday.
