WASHINGTON — Democrats are vowing to keep up the fight for an expansion of government-funded health insurance after they failed to override a presidential veto of a bill that would have added nearly 4 million children to the rolls.

Yesterday’s House tally, which fell 13 votes short of the two-thirds majority needed to override, was widely expected, but it nonetheless provoked a torrent of outraged responses from Democrats at once frustrated by their inability to persuade enough Republicans to oppose President Bush and eager to seize what they view as a political opening.

While the White House declared victory and urged a compromise, the Democratic leadership tried to signal defiance. “In the coming days, Democrats will not back down, and we will insist on providing health care coverage to these 10 million children,” the chairman of the House Democratic caucus, Rep. Rahm Emanuel, said. The House speaker, Rep. Nancy Pelosi, indicated that another bill would be sent to the president within two weeks.

Yet that attitude did not appear to be universal among Democrats. The party’s presidential frontrunner, Senator Clinton, said at a health care forum yesterday that the sweeping reforms that she and other Democrats want would have to wait until a new presidential administration and a larger majority for the party in Congress.

“We’ll work to see what kind of compromise might be available between this White House and the Congress,” she said, referring to the health program known as SCHIP. The bill provided an annual spending increase of $35 billion for SCHIP, which would have expanded to include 10 million children from its current 6 million. The increase would be funded by raising the cigarette tax.

Mr. Bush has proposed a $5 billion, or 20%, increase in funding to reauthorize the program. In rejecting the legislation originally passed by Congress he cited his opposition to increasing taxes and said the expansion would provide government health insurance to families that don’t need it, including some that earn up to $83,000 a year. He also said it was an incremental step on the road to government-run health care.

Looking to claim the high ground in the political battle, the president this week designated three senior officials to negotiate with Congress, including the secretary of health and human services, Michael Leavitt. He has also indicated that he would be willing to increase funding to cover 500,000 children who are now eligible for SCHIP, but not enrolled. “As it is clear that this legislation lacks sufficient support to become law, now is the time for Congress to stop playing politics and to join the president in finding common ground to reauthorize this vital program,” the White House said in a statement. The president’s chief spokeswoman said at a press briefing: “We won this round on SCHIP.”

Democrats clearly see it differently. Even as they express disappointment at their failure to pick up enough Republican votes to override, they view the fight over children’s health care as a winning political issue for 2008, citing polls showing the bill has overwhelming popular support.

They have ridiculed Mr. Bush and Republicans in Congress for citing fiscal responsibility in opposing the legislation, saying the claim rings false from a party whose fiscal record has drawn criticism in recent months from several of its presidential candidates and from the former chairman of the Federal Reserve, Alan Greenspan.

The dean of New York’s congressional delegation and the Democratic chairman of the House ways and Means Committee, Rep. Charles Rangel of Harlem, professed on Wednesday to be astonished that the president would leave his party so politically vulnerable by demanding that GOP lawmakers vote against health care for children. “What he’s doing to Republicans even I would have second thoughts on,” Mr. Rangel told reporters.