Senator Clinton is beginning her historic bid for the White House by returning to the issue that bedeviled her when she first resided there: health care.

A day after announcing her candidacy for president, Mrs. Clinton appeared at a West Side health clinic to push legislation that would expand a federal program that subsidizes health coverage for children of low- and moderate-income families.

The former first lady, surrounded by local elected officials and clasping the hand of a small girl, said her proposal would provide “quality, affordable health care” to every child in the country. Universal health care, she said, would play “a very major part” in her presidential campaign.

“We start from the premise it is simply wrong for any child in America to go without health insurance. That’s where we start,” she said. “If you disagree with that, then obviously you don’t share our values and our views that in the richest of all countries we have both the obligation and now the opportunity to make sure no child does go without health insurance.” Mrs. Clinton, who led an unsuccessful effort to enact universal health coverage in 1994, stood on a stage at the Ryan Chelsea-Clinton Community Health Center, named not for her daughter, Chelsea Clinton, but for the late Rep. William Fitts Ryan, a Democrat of New York, and for the Manhattan neighborhoods adjacent to its location, on Tenth Avenue between 45th and 46th streets.

Her proposal, which she said will be introduced jointly in the Senate and House, would expand the State Children’s Health Insurance Program to allow eligibility for children of families making 375% of the poverty level, which in New York is now $75,000 for a family of four. Currently, the threshold is 250%, which amounts to $52,000 for a family of four. The federal program is administered in New York State through Child Health Plus.

The bill comes as President Bush is proposing a plan to make health insurance more affordable by changing the tax code to give a break to people buying private insurance while capping the level at which insurance provided by an employer is tax-deductible. The shift, which was published in news reports over the weekend and confirmed by people who have been briefed on the plan by the White House, could lower taxes for Americans whose insurance costs less than the national average and raise the burden for those whose insurance costs more.

“Today the tax code unfairly penalizes people who do not get health insurance through their job,” Mr. Bush said in his weekly radio address. “It unwisely encourages workers to choose expensive, gold-plated plans. The result is that insurance premiums rise, and many Americans cannot afford the coverage they need.” The president said he would detail the plan in his State of the Union speech tomorrow night.

The White House over the weekend reached out to fiscal conservatives to apprise them of the proposal, seeking to pre-empt concerns that Mr. Bush was pushing a tax increase by emphasizing that the plan was revenue neutral.

The president of Americans for Tax Reform, Grover Norquist, called the plan “a fine public policy proposal,” but he said there was “a real danger” that Democrats in control of Congress would turn it into a net tax hike. The White House, he said, “should make it clear that should Congress turn it into a tax increase, then it will be vetoed.”

The president’s plan is unlikely to go far on Capitol Hill, some health policy scholars said yesterday. “I think it will get a very chilly reception from Democrats in Congress,” a former health policy adviser to the president, Doug Badger, said. Mr. Badger is now a senior fellow at the Center for Medicine in the Public Interest. The proposal also could have an impact on the field of Republican presidential candidates, in which Governor Romney is expected to claim leadership on the health care issue. Last year, Mr. Romney signed legislation making Massachusetts the first state in the nation to require that every citizen have insurance.

Mr. Badger said the president’s federal plan could complement the Massachusetts program, since it would provide more aid for the state to use in subsidizing health coverage for its residents. But Mr. Romney, he said, might also distance himself from the part of Mr. Bush’s proposal that would increase the tax burden for individuals whose insurance costs more than the national average.

A spokesman for Mr. Romney did not respond to requests for comment yesterday.

In New York, Mrs. Clinton also did not address the president’s plan. Her event, which as many as 100 journalists attended, was tantamount to her first campaign stop, as photographers snapped countless pictures of her playing with 3-year-old Camilla Harden and her sister Olivia, who turns 2 in March.

Calling President and Chelsea Clinton her greatest advisers, Mrs. Clinton said she was running because she was the candidate “best positioned” to meet the challenges America faces. Her campaign team has already been trying to hammer home that point, yesterday releasing a memo by Mrs. Clinton’s pollster, Mark Penn, highlighting her high standing among Republican and Democratic candidates in national polls. The memo did not mention some state polls in Iowa, which have shown her trailing Senator Edwards and running neck-and-neck with Senator Obama of Illinois and Governor Vilsack of Iowa.

Mrs. Clinton is scheduled to appear today at ground zero along with other members of the city’s congressional delegation to push for more federal aid for responders to the September 11, 2001, terrorist attacks on the World Trade Center.