In the session at the White House, when President Bush reported on progress of the war, governors pressed him to provide more money so they could guarantee health insurance for children. In response, administration officials said states should make better use of the money they already had.

Gov. Sonny Perdue of Georgia, a Republican, said afterward, “Health care for children ought to be a priority, irrespective of anyone’s views on the war.”

Georgia will exhaust its allotment of federal money for the Children’s Health Insurance Program within three months, Mr. Perdue said. Thirteen other states expect to run out by September, according to data released here at the winter meeting of the National Governors Association.

Governors said the Clinton and Bush administrations had encouraged them to expand children’s coverage and had granted waivers allowing them to cover parents and even some childless adults.

Having successfully expanded the health insurance programs in their states, some governors now suggest that the Bush administration is pulling the safety net out from under many children.

In his budget this month, Mr. Bush said he wanted to return the program to its “original objective” of covering children with family incomes less than twice the poverty level. Budget documents note that 16 states cover children above that level and that “one state, New Jersey, covers children up to 350 percent of the federal poverty level.”

A family of four is classified poor if its annual income is less than $20,650.

An influential member of Congress said Monday that he would not be taking up White House proposals to restrict eligibility and financing for the child health program.

“I have absolutely no intention of moving the president’s proposals through our subcommittee,” said the lawmaker, Representative Frank Pallone Jr., Democrat of New Jersey.

Mr. Pallone is chairman of the Health Subcommittee of the Energy and Commerce Committee, which has authority over the children’s program.

Speaker Nancy Pelosi said Monday that “Democrats in Congress understand the urgency” of the problem and would provide money to the 14 states that did not have enough to cover their current enrollment. Although Mr. Bush would reduce federal payments for adults and for children with family incomes above 200 percent of the poverty level, Mr. Pallone said states should have discretion to cover children above 200 percent of the poverty level and adults in some circumstances, too.

“In New Jersey, we made a decision to go up to 350 percent of the poverty level, because we have the highest cost of living in the country,” Mr. Pallone said.

Likewise, he said, New Jersey found that covering adults increased the likelihood that their children would stay on the rolls.

“The hallmark of all this is flexibility,” Mr. Pallone said. “A robust Children’s Health Insurance Program is an important part of any effort to try to achieve universal coverage.”

The federal government spends $5 billion a year on the program. Mr. Bush wants to continue that level, and he is seeking an ”additional allotment” of $4.8 billion over the next five years.

States would need substantially more to continue their programs with current eligibility rules and benefits. New estimates from the Congressional Budget Office show that the states face shortfalls of $700 million this year and a total shortage of $13.4 billion from 2008 to 2012.

Gov. Jim Douglas of Vermont, a Republican, said the Bush proposals would jeopardize his state’s phenomenal success in covering children. In Vermont, he said, fewer than 4 percent of the children are uninsured, and “we don’t want to lose ground.”

Bush administration officials emphasized that states received a fixed amount of federal money each year, and they said individual children did not have a legal entitlement to benefits. Michael O. Leavitt, secretary of health and human services, said he would work with Congress to find “a short-term solution” for states exhausting their allotments this year. He said states could avoid shortfalls by managing their programs better.

In his experience as governor of Utah, Mr. Leavitt said, “when we were out of an allotment, we just discontinued enrolling people until we had room.” Likewise, he said, states could cover more people if they provided less comprehensive benefits.

Gov. Ted Strickland of Ohio, a Democrat, said: “If we don’t get the money we need, children will go without coverage.”

“In the meeting with the president and Secretary Leavitt,” Mr. Strickland said, “when questions were raised about children maybe having to be removed from the program or eligible children not being able to participate, we were told that that was basically a management problem.”

Gov. Jon Corzine of New Jersey, a Democrat, said that under the president’s proposals “we will end up having fewer children covered.” That prospect “was chilling to some of us,” Mr. Corzine said, adding that states wanted to avoid “rationing health care to our most vulnerable and our most needy.”

Gov. Edward G. Rendell of Pennsylvania, a Democrat, said Mr. Bush’s budget request was “clearly insufficient” to continue coverage for the six million children enrolled in the program.

Many governors want to expand the program, which they see as a foundation for their efforts to expand coverage generally.

Mr. Rendell framed the issue as a choice, asking: “Should we be giving tax cuts to billionaires and millionaires or should we be giving health care to children? Should we make health care for children, at the very least, an entitlement?”

Domestic policy is in a straitjacket because of the cost of the war, the cost of tax cuts and the president’s plan to balance the budget within five years, Mr. Rendell said.

Gov. Arnold Schwarzenegger of California, a Republican, said federal aid was essential to his $12 billion plan for universal health coverage. Mr. Schwarzenegger said that in a private meeting he told the president, “We need the federal government’s help.” He did not say whether he got a commitment.