President Bush encouraged governors Monday to support his call for changing the tax code to help more people buy private health care insurance, but did not address their pleas to increase funding for a health care program that insures millions of children of the working poor.

Still, governors said they heard words of at least partial compromise from the administration on a budget dispute that dominated private discussions among governors Sunday.

At stake is coverage for 6 million people, overwhelmingly children, as well as the hopes of many governors in tackling the larger challenge of the uninsured. All governors rely on the State Children's Health Insurance Program, intended to aid uninsured working families.

Bush, welcoming the governors after they met privately with several administration officials, did not offer any comments about the children's health program, talking rather about his larger proposals.

"I'm looking forward to working with Congress on health care. I firmly believe ... that states are often times the best place to reform systems and work on programs that meet needs," he said.

But Health and Human Services Secretary Mike Leavitt "made it clear that the administration will work with Congress as far as" short-term shortfalls, said North Dakota Gov. John Hoeven, a Republican. Governors say 14 states could run out of cash before October. In Georgia, it could happen as soon as March.

The governors want two things:

—Enough money to keep the program afloat through October. That is estimated at $745 million.

—Changes to Bush's budget. Analysts say his spending plan would shortchange the health program even if the number of people served did not grow. The long-term shortfall is put at $10 billion to $15 billion over the next five years.

Gov. Jon Corzine, a New Jersey Democrat, warned that the administration's budget promised illusory savings. "You end up paying for this in other ways — uncompensated care, emergency rooms," Corzine said. "This is pay me now or pay me later."

Corzine said he still wanted more clarity from administration officials on support for the short-term funding, but said Leavitt had offered words of compromise.

But the long-term issues over the program remained in dispute, governors said.

The program, approved in 1997, covers uninsured children whose families earn too much to fall under Medicaid, the joint state-federal health care service for the poor.

More than a dozen states have expanded SCHIP, with consent of the federal government, to cover adults in those families. The program now insures an estimated 639,000 adults among its 6 million.

Many governors said the administration's efforts to scale back the program would undermine state efforts to craft universal health care plans. Many of these have started with a target of insuring all children.

California, Massachusetts and Pennsylvania have developed some of the most ambitious proposals to try to get to universal health care coverage. Most states have just tried to strengthen their health care system to cover more people.

At their private session Sunday, governors said there was bipartisan support for help on the immediate needs and a long-term commitment to the current program.

Leavitt said Sunday that there is enough money among states to cover short-term shortfalls, if states with surpluses would share with those with deficits, an idea that has little support among governors. Bush wants SCHIP to remain focused on poor children, not all children and not adults, Leavitt said.

Gov. Ed Rendell, D-Pa., said he was confident that a compromise on the money can be found. He said the administration has been helpful to his efforts to expand coverage and approved a waiver that would let the state cover 180,000 more children. "I want to give the administration high praise," he said.

But most were worried. In Rhode Island, GOP Gov. Don Carcieri said aggressive enrollment efforts had boosted their combined Medicaid and SCHIP program so that 94 percent of children were covered, at its height, before administrative hurdles and other problems caused some backsliding.

"We built all that up," Carcieri said. "We don't want to pull the rug out."