Sullivan: West Ham have bid for four strikers
Manager Sam Allardyce was tasked with improving the Upton Park side's style of play after protestations from supporters last season, and has already added two strikers to his ranks in the form of Enner Valencia and Mauro Zarate.
But, having been rocked by news of Andy Carroll's ankle injury, which will rule the club record signing out for around four months, Sullivan acknowledges the need to add extra firepower to the squad.
"We have got offers in for three or four strikers - one in England, three abroad - and we just hope to land one of them," he told the club's official website. "We do need to sign another player.
"We have worked 24/7 all summer and we still want to get at least one more striker in. It's been very, very tough but we're pleased with what we've got - there are players with massive potential for the future but we know there isn't enough goals in the team at the moment."
On the subject of Carroll, who has suffered a catalogue of injuries since moving to Upton Park, Sullivan could not hide his frustration.
"The pre-season hasn't gone quite as planned - the worst thing is that we picked up injuries and in particular Andy Carroll, who is out for four months and makes a huge difference to us.
"We've got to overcome these things and, if you look at the history of football, often people with a good pre-season have a terrible season, and people who have a bad pre-season have a good season."
He was, however, more upbeat about the potential of Ecuador striker Valencia, a reported £12 million signing from Mexican outfit Pachuca.
"The player we've spent the most money on, Enner Valencia, is our big summer signing and he is about two or three weeks away from being able to play, because he was at the World Cup.
"He might be in the top five scorers in the Premier League next season, he might take time to adjust, you just don't know with foreign players, but potentially he's a top, top player."
West Ham have one more pre-season friendly, at home to Serie A side Sampdoria on Saturday, before beginning their Premier League campaign at home to London rivals Tottenham on August 16.