The Hammers attempted to sign Carroll on a loan deal with an option to buy and even had an agreement in place with the Reds, but the striker did not want to leave Anfield for Upton Park.
"I can only talk about this because it has been put in the public domain already," Sullivan told the Daily Mirror.
"My manager wanted Andy Carroll, so we backed him. Andy was the perfect player for him and the style of football we play.
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"We play a lot of high balls, a lot of crosses - quite exciting football, not just humping it but clever crossing - and Sam wanted [Wolves winger Matt] Jarvis to supply the crosses because he is a fantastic crosser.
"We had a deal in place with Liverpool to borrow the player which automatically became a purchase if we stayed up.
"We hoped, as he had the same agent as our manager - a man we had done a lot of business with - and he was very friendly with Kevin Nolan, our captain, that Andy would want to come.
"But he really wants to stay at Liverpool. He believes that he will get back into the side by his performances - coming off the bench or for the reserves. He believes the manager will have to put him back in the side.
"But it's wrong to discuss what is in his mind. Only he knows that."
Newcastle United have also expressed interest in their former player, but Sullivan says the financial package the Magpies offered was not tempting enough for Liverpool.
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"That is what we are led to believe. So Liverpool were not prepared to let him go to Newcastle. But it may all change. Who knows?
"It was a terribly expensive deal with the loan fee and his wages, which were enormous - too much for a club like ours.
"But, as the manager really wanted him, we were prepared to pay it. Then, there was a huge transfer fee which would have blown our budget for several years. So, as we buy players, the door shuts."
One of the players the Hammers have since acquired is France international Alou Diarra, who Sullivan claims could be one of the best purchases this summer. The midfielder was previously on Liverpool's books in 2002, but did not make a first team appearance before leaving again three years later.
"[Diarra] was a boy at Liverpool," he explained. "A kid. He had not come to his strength and he didn't play a game for Liverpool.
"But he is a supreme professional. I would be very surprised if he doesn't turn out to maybe be one of the buys of the summer."