"Everybody wants children to have health insurance," Health and Human Services Secretary Mike Leavitt said. "The president has already said, 'I want a compromise.'"
Mr. Leavitt added, "If it takes more money, we'll put it up." He did not cite a figure.
The legislation passed last week would increase spending for SCHIP by $35 billion over five years. Mr. Bush has called for a $5 billion increase.
In both the House and Senate several Republicans have sided with majority Democrats on the issue, and the Senate has enough votes to override the veto. But in the House, Speaker Nancy Pelosi has delayed an override attempt until Oct. 18, saying she needs to attract support from 14 more Republicans.
"It's hard to imagine how we could diminish the number of children who are covered," said Ms. Pelosi (D., Calif.) in an interview broadcast 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.
Messrs. Bush and 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."
But in the wake of a threatened veto override, the Bush administration offered fresh promises it would seek more money for the program if a need is shown.
"If covering poor children requires more than the 20% increase the president proposed, he's committed to working with Congress to find the additional money," White House spokesman Tony Fratto said Sunday.