The 23-year-old has scored once in 11 Premier League appearances for the west London side after moving on a season-long loan to Upton Park from the Reds in August.
And Sullivan says West Ham have agreed a deal with the Reds to buy the striker at the end of the season, but the Hammers are yet to reach an agreement with Carroll.
“West Ham have an agreement with Liverpool to sign him at the end of the season, but we have no agreement with the player,” Sullivan told the Daily Mail.
Carroll has been hampered by injuries since joining Sam Allardyce’s side, limiting the striker’s impact at West Ham.
However, Sullivan believes Carroll is desperate to prove his worth at the Hammers, with 15 Premier League games remaining this term.
“It was his desire to come to West Ham to prove himself to Liverpool and that hasn’t changed,” added Sullivan.
“He has had injuries since he came but is back to fitness now and we have 15 games left. We need some goals and some points so let’s see.”
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