WASHINGTON — President Bush’s health secretary said yesterday he does not expect Congress to override a veto on children’s insurance and warned that the popular program could be at risk unless Democrats restrain spending.

In an interview with the Associated Press, Health and Human Services Secretary Mike Leavitt said Mr. Bush would be willing to provide more than the $5 billion increase over five years that he first proposed. He declined to say how much additional money was possible. But in a warning to Democratic leaders who have pledged to stick with their $35 billion increase, Mr. Leavitt said Mr. Bush would not waver despite attempts to override his veto last week.

An override requires a two-thirds majority in the House and Senate. The Senate approved the increase by a veto-proof margin, but the House fell about two dozen votes short of a two-thirds majority. The House has scheduled an override vote for October 18.

Mr. Leavitt said the Democratic-controlled Congress, not the Republican administration, would pay the political price if the State Children’s Health Insurance Program stalls due to gridlock. Congress has continued funding the program at its current level until mid-November as part of legislation keeping government agencies operating beyond October 1, the start of the new budget year.

“I’m presuming the Democrats do in fact want the children’s insurance program to be reauthorized,” Mr. Leavitt said.

“The president knows bad policy when he sees it. He has said as clearly as possible that ‘I want to reauthorize this program and I’m prepared to add to the 20% increase I’ve already proposed.’ But we need to have a serious conversation that involves all of the points of view,” Mr. Leavitt said. He added, “Once we agree on our priorities, then the proper number will arrive.”

After his veto, Mr. Bush immediately signaled a willingness to compromise on a new bill, but congressional Democrats stood firm. “You cannot wring another ounce of compromise out of this,” the Senate majority leader, Harry Reid, a Democrat of Nevada, said last week.

The House speaker, Rep. Nancy Pelosi, is seeking support from 14 more Republicans to vote against the GOP president.

“It’s hard to imagine how we could diminish the number of children who are covered,” Ms. Pelosi, a Democrat of California, said in an interview broadcast on “Fox News Sunday.” “The president calls himself ‘the decider,’ and I don’t know why he would want to decide that one child has health care and another does not.”

“So we take it one step at a time. And right now, we have the next 10 days to two weeks to try to peel off about 14 votes in the House,” she said. The program provides health insurance to children in families with incomes too great for Medicaid eligibility but not enough to afford private insurance.

Mr. Bush and Mr. Leavitt have decried the spending increase primarily supported by Democrats as unnecessarily subsidizing middle-income people as part of Democrats’ “goal of government-run health care for every American.” Mr. Leavitt said the veto override effort was sure to fail and as a result would waste two weeks during which the administration could work with Congress on new legislation.