With Andy Carroll sidelined for four months due to an ankle injury, Sullivan is looking to bring in another striker to provide manager Sam Allardyce with more options.
The Hammers have already made six signings this window, but are looking to further add to their squad.
Sullivan told the club's website: "We've got offers in for three or four strikers - one in England, three abroad - and we just hope to land one of them. 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."
West Ham are looking to bring in one more striker to join Enner Valencia and Mauro Zarate in supplementing the Hammers' frontline for the new season - but Sullivan believes both players will have a positive impact.
"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 added.
"He's now in full training but I think the earliest he'll play is in the Capital One Cup match, so it'll be exciting for people to see him make his debut in that match.
"We were looking at Enner from before the World Cup, when Ecuador played Australia at Crystal Palace and he was the leading goalscorer in Mexico.
"Goalscorers are goalscorers and his goals per game for Ecuador since he's been playing centre forward is very high. He scored more goals at the World Cup than the England football team, he was in the top ten goalscorers of the tournament, even though his team got eliminated at the second stage, so he's a player with potential.
"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.
"If Zarate gets the supply he will score goals. He probably needs to play off a big centre forward but at least we now have a plan B. If we're losing in a game we have someone who can go on and change things."
If we sign Crouch and get Carroll fit, there'll be no need for knock downs - they can just play head tennis and walk it in. Exciting times.- hammerguy