WASHINGTON — Voters can look forward to a preview of what a general election battle between Mayor Giuliani and Senator Clinton would look like if New York lawmakers call Mr. Giuliani or members of his former city administration to appear before an inquiry into the health issues of ground zero workers.

Members of the new Democratic Congress convened the first of what is expected to be several hearings on the topic yesterday as House lawmakers sparred with the Bush administration over a lack of federal funding for key treatment and monitoring programs. Senator Clinton is holding a hearing later this month to examine possible solutions to long-term health effects of working at the World Trade Center site. Mayor Bloomberg agreed to testify at that hearing, to be held through the Health, Education, Labor, and Pensions Committee, headed by Senator Kennedy of Massachusetts.

Mrs. Clinton will lead a subsequent subcommittee hearing that is likely to deal with air quality issues in Lower Manhattan in the more immediate aftermath of the terrorist attacks of September 11, 2001, when Mr. Giuliani was still in office. That hearing has yet to be scheduled. Asked if the senator would seek testimony from Giuliani administration officials, a spokesman for Mrs. Clinton, Philippe Reines, responded that it was “premature” to say.

Other Democrats in the city’s congressional delegation were more certain.

“Absolutely,” Rep. Jerrold Nadler, whose district includes ground zero, said in an interview. Along with Mrs. Clinton, Mr. Nadler has been a chief critic of the federal government’s response to the attacks, accusing officials of misleading New Yorkers about the air quality in Lower Manhattan and failing to clean up the buildings around ground zero. He said he plans to hold his own hearings on the issue.

While lawmakers have focused most of their criticism on Washington, some have questioned whether the Giuliani administration could have done more to protect workers who toiled on the pile amid a cloud of toxic dust in the weeks after the twin towers fell.

“Who made decisions, if any, that resulted unnecessarily in a lot of people getting sick?” Mr. Nadler said.

Referring to Mr. Giuliani, he added: “Clearly things that were done in that time period have to be looked at so we can learn from our mistakes. Whether he should testify or somebody else I don’t know.”

Rep. Edolphus Towns of Brooklyn, who convened yesterday’s hearing on Capitol Hill, said he would hold another one in New York. “Whoever can shed some light on this I think we need to talk to,” he said.

The Giuliani campaign declined comment last night.

Digging into the Giuliani administration’s response could put Mrs. Clinton in the potentially awkward position of interrogating aides to a potential presidential rival. It could also undermine what has been Mr. Giuliani’s chief asset in his own White House bid. “It goes to Giuliani’s narrative of pre- and post-9/11 hero,” a political science professor at Baruch College, Douglas Muzzio, said.

Mr. Muzzio offered this advice to Mrs. Clinton: “Take the high road.”

Taking direct shots at Mr. Giuliani’s leadership would be risky, he said. “It’s not smart strategy. It’s too early.”

At the hearing yesterday, two current deputy mayors, Edward Skyler and Linda Gibbs, urged lawmakers to allocate $1 billion through 2011 for monitoring and treatment programs related to September 11, as well as to transfer another $1 billion to a victims’ compensation fund, so the city will not have to contest lawsuits brought by ground zero workers. The funding request stemmed from recommendations issued earlier this year by a mayoral panel.

President Bush requested $25 million for the programs in his 2008 budget, and an assistant secretary for health and human services, Dr. John Agwunobi, disputed city claims that the current federal funding will run out before the end of this fiscal year.