The 24-year-old Carroll completed a permanent move to Upton Park after spending last season on loan from Liverpool, but with Carlton Cole being released and Marouane Chamakh and Wellington returning to their parent clubs, the Hammers are light of attacking options.
A move for Duvan Zapata failed due to work permit issues but Sullivan, who was conducting a question and answer question through his son's Twitter account, confirmed a new forward remains the target ahead of the new season.
He was asked by fans whether Sam Allardyce was looking to sign a new full-back or attempt to bring Scott Parker back to the club on loan, but the 64-year-old, who now owns over 55 per cent of West Ham, replied to both Tweets stating a frontman was essential.
"We need goal scorers not defenders," he said when questioned about defensive targets, before responding to the possibility of Parker making a return by saying: "We have to focus wages on strikers due to wage cap but would love Scott back if there was room."
When asked if there were other targets, Sullivan reiterated his stance by replying: "We are happy with the rest of the team all we need now is pace and goals."
Sullivan also said loan moves for Chelsea's Romelu Lukaku and Raheem Sterling of Liverpool were unlikely and revealed the former was a target for West Ham some years ago.
But with Carroll nursing a foot injury that could see him miss the start of the season and only Modibo Maiga in reserve, a goalscorer will almost certainly be needed to add to Allardyce's squad if the Hammers are to push on and improve last season's 10th-place Barclays Premier League finish.
- Sports & Recreation
- Sam Allardyce
- Marouane Chamakh
- Scott Parker