The England international, a £35million signing from Newcastle two years ago, made the move from Anfield to Upton Park on a season-long loan deal in the summer.
Carroll's time with the Hammers has been badly hampered by injury and he has managed just one goal from nine Premier League starts and two substitute appearances.
Sullivan admitted that despite reaching an accord with their Premier League rivals, they have yet to agree terms with the powerful striker over a permanent switch and is keen for the 24-year-old to prove his worth over the final 14 games of the Premier League season.
In an exclusive interview with MailOnline, he said: "I am often asked about Andy Carroll's deal. It is a season-long loan. Liverpool had an option to take him back in January, but he has been injured.
"West Ham have an agreement with Liverpool to sign him at the end of the season, but we have no agreement with the player.
"It was his desire to come to West Ham to prove himself to Liverpool and that hasn't changed.
"He has had injuries since he came, but he is back to fitness now and we have 15 games left... we need some goals and some points, so let's see."
Sullivan also revealed how West Ham were keen to do more business during the January transfer window, only to lose out to Leeds in their pursuit of defender Stephen Warnock, while interest in Reto Ziegler and Paul Robinson also came to nothing.
He added: "For all the deals you have read about, there were 30 others we were trying. I thought we had Stephen Warnock from Aston Villa and he said he was signing.
"Our contract offer was for six months with an extension based on appearances and he accepted that. Then he changed his mind and signed for a Championship team in the north.
"We had a deal for the Olympiakos left-back, but when Warnock said yes, we scrapped that. Then, when we tried to get the other deal back on, the chap had gone somewhere else.
"We then spoke to Juventus left-back Reto Ziegler, who's on loan in Russia and tried to make that happen, but he went to Turkey.
"We tried to bring in goalkeeper Paul Robinson from Blackburn, but he had an injury and hasn't played often and we had a bit of doubt."
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