Hull have been unsuccessful in their quest to sign a striker this summer with Marc Antoine Fortune opting to join Celtic and Fraizer Campbell still stalling on a move to the KC Stadium, despite the club agreeing a fee with Manchester United early last month.
Brown has since turned his attention to Zamora, who scored twice for Fulham in 32 league games last season.
"We've been active with regards to Fulham," Brown said. "There is a firm and substantial bid on the table for Bobby Zamora.
"I'm waiting to hear from my chairman for the next step.
"There are substantial bids on the table for four players, one being Fraizer Campbell and the other being Bobby Zamora."
Hull retained their Premier League status on the final day of last season after a dramatic slump in form after Christmas.
But Brown is confident the club will do better next term.
"There are a number of players out there that we want to bring to this football club," he said.
"We're pitching it high this year rather than being content with staying in the division like last season.
"Hopefully the players that come through the door will reflect that.
"There are a number of deals ongoing but am I anxious or nervous that people have not arrived yet? No I'm not.
"There's a lot of hard work going on in the background and it's constant effort going in to getting stronger individuals to improve the team."