WASHINGTON — President Bush indicated Saturday that he would be willing to accept a larger increase for a children's health insurance program than the one he has proposed but defended his veto of the expansion of coverage approved by Congress.

Bush's long-promised veto Wednesday set off an ideological battle about who holds responsibility for extending health-care benefits to uninsured children: the government or the private sector.

The congressional bill would spend $60 billion over five years to expand health coverage for children of the working poor and middle-class, and it would pay for it with higher tobacco taxes. Bush has offered $30 billion, a 20 percent increase over current levels but not enough to maintain the existing enrollment in what is known as the State Children's Health Insurance Program, budget analysts say.

The program is managed by states within federal guidelines and serves about 6 million children. An estimated 9 million children remain uninsured in the United States, and the number has been rising as employers cut back coverage.

Bush's veto led one Democratic lawmaker last week to call the president "Ebenezer Scrooge," while a Republican pollster noted that "it will take some superb communications to persuade voters that the White House really is on the side of children's health."

During his weekly radio address Saturday, Bush called for a compromise but offered no specifics.

"If putting poor children first takes a little more than the 20 percent increase I have proposed in my budget for SCHIP, I am willing to work with leaders in Congress to find the additional money," he said.

Bush earlier hinted he was open to a compromise but still has not made clear what he is willing to accept. He continued to describe the measure that he vetoed as "deeply flawed," contending that the plan was "an incremental step toward their goal of government-run health care for every American," which he believes is "the wrong direction for our country."

Rep. Steny Hoyer, a Maryland Democrat who is the House majority leader, pointed out that most children enrolled in SCHIP receive coverage through private insurers who hold state contracts, even though the government subsidizes the benefits.

"The truth is, America's largest private insurance lobbying group supports this bill—as do America's doctors, nurses, children's advocates and, most importantly 72 percent of Americans," Hoyer said in the Democrats' response to Bush's address.

The current law, which remains in effect while the debate over reauthorizing it continues, covers children in families earning up to $40,000 a year, about twice the federal poverty level. But some states obtained permission to extend eligibility to families with higher incomes, and the bill would authorize states to allow households with an income of about $60,000 a year to enroll their children in SCHIP.

Bush also noted that six states project that they will spend more SCHIP money on adults than they do on children in this fiscal year. However, those states had federal permission, in many instances during the time Bush has been in office, to cover adults. The president urged both parties to come together to support a bill "that moves adults off this children's program."