NEW YORK - Ailing ground zero workers went to court Tuesday to demand that the company overseeing a $1 billion Sept. 11 insurance fund spend the money to pay for their health care.

The workers have already filed a class-action lawsuit claiming the toxic dust from the World Trade Center site gave them serious, sometimes fatal diseases. On Tuesday, they sought compensation from the WTC Captive Insurance Co., the company in charge of money appropriated by Congress to deal with Sept. 11 health-related claims.

“The WTC Captive has consistently refused to pay any of the ground zero workers who have become ill on the work site, including any compensation” for lost salaries, pain and suffering, medical treatment, medical monitoring or burial expenses, according to the lawsuit, filed in state Supreme Court in Manhattan.

It was filed by attorneys representing thousands who became ill after working to clean up the site while breathing toxic trade center dust, including more than 100 who have died.

“She hasn’t paid a penny to one of my 10,000 people,” David Worby, an attorney representing the workers, said of the company’s CEO, Christine LaSala. “It was their mandate.”

Mayor Michael Bloomberg, who was named in the suit along with LaSala and board members, said Tuesday that attorneys are wrong about the company’s structure.

“They just don’t know the facts. The truth of the matter is, Congress didn’t set up a victims’ compensation fund,” the mayor said. “We’d like them to do that, we’ve asked for that; they set up a captive insurance company. And the insurance company can only pay out monies if somebody sues us in court and wins a judgment against us.”

The lawsuit, relying on testimony from federal officials over the years about the fund’s purpose, said officials meant for the money to be used to compensate ailing workers.

The company said in a statement Tuesday that the lawsuit was “completely without merit” and that WTC Captive has been operating as the federal government intended it, insuring the city and its contractors from Sept. 11 claims.

Since it began operating in 2004, the company has spent more than $73 million of the insurance money in legal fees and other expenses, the lawsuit says.

Worby and other attorneys plan to go to Washington this week to lobby congressional leaders to urge the company to make the $1 billion available immediately for sick workers.

Bloomberg and other city officials have estimated the cost of caring for the workers who are sick or who could become sick at $393 million a year and urged the federal government to pay for their treatment and monitoring.