Senate Majority Leader Harry Reid decried Mr. Bush's action as a "heartless veto."
"Never has it been clearer how detached President Bush is from the priorities of the American people," Reid, D-Nev., said in a statement. "By vetoing a bipartisan bill to renew the successful Children's Health Insurance Program, President Bush is denying health care to millions of low-income kids in America."
Democratic congressional leaders said they may put off the override attempt for as long as two weeks to maximize pressure on Republican House members whose votes will be critical.
"The President should not be so heartless when it comes to the children of America," Washington State Democrat Maria Cantwell said on the floor of the Senate. "I know my colleagues here are working shoulder to shoulder, Democrats and Republicans, trying to stop the President's veto."
"We remain committed to making SCHIP into law - with or without the president's support," said the leader of the House, Speaker Nancy Pelosi, a Democrat, referring to the full name of the State Children's Health Insurance Program.
Eighteen Republicans joined Democrats in the Senate, enough to override Mr. Bush's veto. But this was not the case in the House, where despite sizable Republican backing, supporters of the bill are about two dozen votes short of a successful override.
Illinois Rep. Rahm Emmanuel talked to CBS News chief White House correspondent Jim Axelrod about the political advantage the Democratic party sees in this situation.
"So the President of the United States is asking 15 Republicans to stand with him on an argument about government-run health care… and deny American kids health care," Emmanuel said, "and yet, vote at the same time to give Iraq 190 billion dollars."
House Majority Leader Steny Hoyer, D-Md., said Democrats were imploring 15 House Republicans to switch positions but had received no agreements so far.
House Minority Whip Roy Blunt, R-Mo., said he was "absolutely confident" that the House would be able to sustain Mr. Bush's veto.
Senate Minority Whip Trent Lott, R-Miss., said Congress should be able to reach a compromise with Mr. Bush once he vetoes the bill. "We should not allow it to be expanded to higher and higher income levels, and to adults. This is about poor children," he said. "But we can work it out."
The White House sought little attention for Mr. Bush's action, with the president casting his veto behind closed doors without any fanfare or news coverage. He defended it later Wednesday during a budget speech.
"Poor kids first," Mr. Bush said. "Secondly, I believe in private medicine, not the federal government running the health care system."
But he seemed eager to avert a full-scale showdown over the difficult issue, offering that he is "more than willing" to negotiate with lawmakers "if they need a little more money in the bill to help us meet the objective of getting help for poor children."