Injuries have restricted the 24-year-old striker to just nine Premier League starts and two substitute appearances for Sam Allardyce’s side since arriving from Anfield on a season-long loan deal in the summer.
Despite that Hammers co-owner Sullivan revealed an agreement is in place for Carroll, who joined the Reds for a record £35 million from Newcastle two years ago, to remain at Upton Park beyond the current season, although the player has yet to agree terms on the switch.
“I am often asked about Andy Carroll’s deal,” he said. “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,” Sullivan added.
“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.”