WASHINGTON (CNN) -- Congress and President Bush are headed to a showdown with no sign of compromise on a popular children's health care bill .
Musician Paul Simon, left, joins House Speaker Nancy Pelosi to oppose Bush's veto of the SCHIP bill.
The House of Representatives is scheduled to vote Thursday on whether to override Bush's veto of a plan expanding the state-run Children's Health Insurance Program.
Though Democrats have pounded Republicans over the issue for two weeks, House GOP leaders predict they will have the votes needed to uphold the veto.
But House Speaker Nancy Pelosi told reporters Wednesday, "We are still in this fight."
"The president is alone, and he's dragging some of his House members with him down this path," the California Democrat said.
Pelosi vowed Democrats will keep pushing the proposed five-year expansion even if they fail to override Bush's veto.
She was also adamant the plan -- which would provide $35 billion for children's health care and which supporters say will cover 10 million children -- would not be scaled back.
"Oh no, 10 million children, we won't scale back one child. Ten million children, thank you," Pelosi said. Watch Pelosi, kids, rock star make plea for plan »
"No. No. No," Senate Majority Leader Harry Reid said this week when asked if he was open to a deal on the measure that would provide $35 billion for children's health care.
The State Children's Health Insurance program matches money from the states to provide health insurance to children in families with incomes too great for Medicaid eligibility, but not enough to afford private insurance.
It currently covers about 6 million children from families with incomes up to 200 percent of the poverty level, or a yearly income of about $41,000 for a family of four under current national guidelines.
The bill the House and Senate passed in September would extend eligibility for the program to about 4 million more children, paying for the expansion with a 61-cent-per-pack increase in the federal tax on cigarettes.
Critics have said their concern is that parents might be prompted to drop private coverage for their children to get cheaper coverage under the bill.
Bush said Wednesday that the congressional plan would "encourage people to move from private medicine to the public."
At a news conference, the president said the override attempt would fail. He called for compromise, saying he would support a bill that provided enough money to cover half a million children who aren't covered now. That would amount to $5 billion, the president says.
"I want to provide enough money to make sure those 500,000 do get covered. That ought to be the focus of our efforts," Bush said.
But Reid had said Democrats had gone as far as they would go on a compromise, having come down from an amount of $70 billion in the original House bill.
"We have compromised and compromised and compromised. For the president to come now and say, 'Let's compromise,' is disingenuous," Reid said.
"We have squeezed all the juice out of this that we can, and the only thing we're going to agree to is wind up having these 10 million very needy children covered," Reid said.
Congress sent the legislation to the White House on October 2 after the Senate voted 67-29 a week earlier to expand the current SCHIP program.
Though 67 votes in the 100-person Senate would suffice to override a veto, the 265-159 House vote on September 25 is short of the two-thirds majority needed.
Forty-five Republicans voted with the Democrats on September 25 to reauthorize the program and direct $35 billion over the next five years to states to cover children's health-care costs. Eight Democrats voted against the bill.
A new CNN/Opinion Research Corporation poll shows strong public support for a veto override.
The poll suggests 61 percent of Americans want Congress to do it, while 35 percent do not.
"The bill that Bush vetoed appears to be nearly as popular among moderates as liberals," said CNN polling director Keating Holland. "Conservatives are in the president's camp with only 43 percent support overriding the veto."
Bush said raising the income eligibility threshold in the bill "is an attempt by some in Congress to expand the reach of the federal government in medicine."
"I believe in private medicine, not the federal government running the health care system," Bush said after vetoing the bill. He said the government's policy should be to help people find private insurance. E-mail to a friend