WASHINGTON — President Bush cast a quiet veto yesterday against a politically attractive expansion of children’s health insurance, triggering a struggle with the Democratic-controlled Congress certain to reverberate into the 2008 elections.

“Congress will fight hard to override President Bush’s heartless veto,” the Senate majority leader, Harry Reid of Nevada, said.

Republican leaders expressed confidencetheyhaveenoughvotes to make the veto stick in the House, and not a single senior Democrat disputed them. A twothirds majority in both houses of Congress is required to override a veto.

Mr. Bush vetoed the bill in private, absent the television cameras and other coverage that normally attend even routine presidential actions. The measure called for adding an estimated 4 million mostly lower-income children to a program that currently covers 6.6 million. Funds for the expansion would come from higher tobacco taxes, including a 61-cent increase on a pack of cigarettes.

“Poor kids first,” Mr. Bush said later in explaining his decision, reflecting a concern that some of the bill’s benefits would go to families at higher incomes. “Secondly, I believe in private medicine, not the federal government running the health care system,” he added in remarks to an audience in Lancaster, Pa.

The president said he is willing to compromise with Congress “if they need a little more money in the bill to help us meet the objective of getting help for poor children.”

It was the fourth veto of Mr. Bush’s presidency, at a time his popularity is low, the legislation popular enough to draw support from dozens of GOP lawmakers, and an override certain to seal his lame-duck status.

Democratic leaders scheduled the showdown for October 18 to allow two weeks for pressure to build on Republicans. A union-led organization said it would spend more than $3 million trying to influence the outcome. “It’s going to be a hard vote for Republicans,” the House speaker, Rep. Nancy Pelosi, a Democrat of California, said. Criticism of the veto was instantaneous, from every quarter of the Democratic political firmament.

Senator Biden of Delaware, a presidential hopeful, called it unconscionable, party chairman Howard Dean labeled it appalling, and Ms. Pelosi said, “It’s very sad that the president has chosen to veto a bill that would provide health care for 10 million American children for the next five years.”

Republicans said none of the criticism would matter. “I’m confident that the more time we have to explain the veto, the more people will be with their position,” Rep. Roy Blunt of Missouri, second-ranking GOP leader in the House, said.

Longer term, Republicans said their goal was to sustain the veto and force Democrats into negotiations on a compromise that GOP lawmakers could embrace.

The original Senate vote was 67–29, enough to override. But the House votes first, and if Mr. Bush’s allies sustain his veto there, the bill dies.