Representative John R. Kuhl Jr. of New York received just his second telephone call ever from his state's Democratic governor, Eliot Spitzer, last week and was not surprised at the topic: children's health insurance.

"He said, 'I am calling you to come over to the dark side,' " said Mr. Kuhl, who was urged by the governor to drop his opposition to health care legislation and join the effort to override President Bush's veto of the bill.

Mr. Kuhl, a Republican who narrowly survived the Democratic sweep of 2006, said he was unlikely to budge. As a result, voters in his district will also be getting calls - from Democrats and advocacy groups who are planning a telephone, radio, television and even text-message barrage against Republicans over what is shaping up as a defining domestic policy issue of the 2008 campaign.

Democrats believe they have Republicans - short on campaign cash, contending with a spurt of retirements and quarreling - on the run over the legislation, the State Children's Health Insurance Program. Party leaders say the willingness of so many House Republicans to stick with Mr. Bush in the face of bipartisan backing for a $35 billion expansion of the program to provide insurance for poor children will prove costly as Election Day looms a year from now.

"They know they cannot sustain this vote in the fall of 2008 and they are praying it gets worked out before then," said Representative Rahm Emanuel of Illinois, chairman of the House Democratic Caucus.

The Health and Human Services secretary, Michael O. Leavitt, said Sunday that Mr. Bush was ready to work it out. "The president has already said, 'I want a compromise,' " Mr. Leavitt said on the ABC program "This Week." But Democrats say that they have already compromised with Senate Republicans and they are in no hurry to scale back the plan.

Republicans acknowledge they could suffer some short-term damage from an issue easily framed as either favoring health care for poor children - or not.

"Certainly in the immediate, superficial look, everybody is for covering kids who don't have health insurance," said Representative Adam H. Putnam of Florida, chairman of the House Republican Conference.

But he and other Republicans say they eventually can turn the issue to their advantage by making the case that Democrats are spending too much, taking a first step toward national health care and devoting tax money to coverage for some families who can afford insurance. They contend their stance could have special resonance with conservatives unhappy with the recent Republican reluctance to resist popular spending programs.

"If this was October of next year, I'd be really worried," said Representative Roy Blunt of Missouri, the second-ranking House Republican. "But this is October of this year and the beginning of us getting our credibility back by showing that we are willing to take principled stands on spending."

House Republican leaders are confident they can hold their forces together and sustain the president's veto in a vote scheduled for Oct. 18. But over the next two weeks, Mr. Kuhl and more than two dozen other Republicans will face an onslaught of advertisements and public activities intended to put pressure on them to vote to override it.

The Democratic Congressional Campaign Committee is taking on eight Republicans in competitive districts with a series of automated calls and radio advertisements that remind listeners that their lawmaker gets taxpayer-paid health care while opposing the expansion of the program administered by each state.

Beginning Monday, a coalition of liberal and labor groups will start a $1 million advertising effort, with a national advertisement to run on cable channels and local advertisements aimed at specific lawmakers. The national commercial shows a series of children beginning with a baby girl and states, "George Bush just vetoed Abby." It says Mr. Bush puts excessive war spending over health care at home.

"The president's 'yes men' in Congress need to stand up to Bush and stand up for families who work hard but simply can't afford insurance," said Brad Woodhouse, president of Americans United for Change, one group leading the effort.

The health care fight is coming at an inopportune moment for Congressional Republicans. In the Senate, a string of retirements has created openings for Democrats to increase their slim majority. House Republicans have had retirements of their own and party fund-raising is lagging behind Democrats by a wide margin.

The Republican targets of the advocacy campaign say they do not view it as much of a threat, saying many of their voters will not consider the advertisements credible and that tactics like robocalls can backfire.

"I don't worry about it," said Representative Steve Chabot of Ohio, who noted that he strongly supported the insurance program when it was created in 1997. "I am perfectly satisfied with my vote and there is a range of reasons why I think this is a bad bill."