Andy Carroll has arrived at Upton Park from Liverpool on a permanent deal this summer for a club-record transfer fee, but boss Sam Allardyce remains short of options up front.
Carlton Cole, Marouane Chamakh and Wellington are no longer at West Ham, while Carroll is nursing a foot injury that is likely to see him miss the start of the season.
Sullivan was asked by fans on Twitter whether Allardyce was looking to sign a new full-back or attempt to bring midfielder Scott Parker back to the club on loan from Tottenham.
However, the 64-year-old replied to both questions by stating that a new forward was essential.
"We need goal scorers not defenders," Sullivan 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.