Carroll's future at Liverpool has been uncertain following the appointment of Brendan Rodgers as manager and he has been linked with several rival Premier League clubs.
West Ham were among those to show an interest and a season-long loan deal with a view to a permanent £17million transfer had been lined up by the promoted club.
Sullivan says the proposed deal was 'terribly expensive' for the Hammers, but he was prepared to back Allardyce, who wanted Wolves winger Matt Jarvis to supply the crosses for Carroll.
"I can only talk about this because it has been put in the public domain already," co-owner 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.
"We play a lot of high balls, a lot of crosses - quite exciting football, not just humping it but clever crossing - and Sam wanted 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.
"It was a terribly expensive deal with the loan fee and with 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.
"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."
The England international is reported to favour a return to former club Newcastle United, but Sullivan doubts Carroll will head back to St James' Park.
He added: "We were paying all of his wages and a huge loan fee. Newcastle didn't even want to pay all of his wages.
"This 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?"