The England international has endured a difficult campaign at Upton Park after being allowed to join the Hammers on a season-long loan deal by Reds boss Brendan Rodgers last summer.
Carroll has been blighted by an injury problem and has managed just one goal in 11 Premier League games so far for Sam Allardyce’s side.
However, Sullivan is hoping the player, who cost Liverpool £35million when they signed him from Newcastle, will enjoy an injury-free run in the second half of the season and prove to be a big hit with the east Londoners.
And, if Carroll does hit the goals trail, Sullivan revealed they have already agreed terms on a deal with the Merseysiders to sign him, although personal terms would still need to be ironed out 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.
Sullivan added: “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 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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New Swindon manager Phil Brown revelled in his side's victory at Cambridge in his first game in charge.
Graham Westley believed his side 'disintegrated' in the second half of their 2-0 loss against Wycombe.
Port Vale manager Neil Aspin praised his side's 'fighting spirit' after they came back from two goals down to draw 2-2 with Stevenage at Vale Park.
Garry Monk had every reason to have a wide smile on his face after guiding Birmingham to a vital 3-0 Championship home win over below-par Hull.
Lee Johnson was delighted for Milan Djuric after the striker's 64th minute goal boosted Bristol City's Sky Bet Championship play-off bid.
Sheffield United manager Chris Wilder was left frustrated with his team's quality in the final third after they drew 0-0 against Nottingham Forest.
Mark Robins praised his players resilience as Coventry's game management counted in a hard-fought victory at Crewe.