Sam Allardyce has already added strikers Enner Valencia and Mauro Zarate to his ranks, but the Hammers have been linked with moves for Stoke City's Peter Crouch and free agent Hugo Almeida.
But, following record-signing Andy Carroll being ruled out for four months with an ankle injury, Sullivan has revealed bids are in for one England-based striker and three from abroad.
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"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.
"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."
Despite disappointment with Carrol, Sullivan was 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."