Nine days after President Obama’s $447 billion jobs package was blocked in the U.S. Senate, one of the plan’s key components — which would provide $35 billion to states and local governments to hire teachers and first responders — suffered the same fate late Thursday.

The vote represented the legislative part of a strategy by Democrats to convince voters that they are pushing popular job-creation bills that are being thwarted by Republican opposition.

All 47 Republicans voted against allowing the bill to proceed to a full debate, arguing that temporary stimulus dollars for state and local government would do little to bolster the private sector.

Republicans also opposed imposing a 0.5 percent surtax on million-dollar incomes to pay for the aid, as Democrats proposed. They contended that inclusion of a tax increase signaled that the vote was intended as a campaign tool and was not a serious effort to find bipartisan agreement on spurring job growth.

Two Democrats also opposed proceeding with the measure, as did Sen. Joseph I. Lieberman (I-Conn.); 50 senators voted to move ahead with the bill.

Under the rules of the Senate, 60 votes were needed to continue toward action on the bill.

The vote concluded a week-long whirlwind of rallies and speeches to promote the jobs legislation. Obama barnstormed for the bill on a bus tour through North Carolina and Virginia, and Senate Democrats rallied with teachers and firefighters at the Capitol.

“If they vote against these proposals, if they say no to steps we know that will put people back to work right now, they’re not going to have to answer to me. They’re going to have to answer to you,” Obama told firefighters at a stop in Chesterfield, Va., on Wednesday.

Also failing to advance Thursday was a Republican counter-proposal that resembled a piece of Obama’s plan and would have repealed a 3 percent withholding tax on payments to government vendors set to go into effect in 2013.

Democrats said they could not accept paying for the jobs proposal by reducing spending elsewhere, as the GOP urged. Republicans countered that the opposition proved that Democrats were willing to scuttle parts of Obama’s jobs proposal that enjoyed Republican support to avoid weakening their narrative of GOP obstinacy.

Democrats plan to hold votes on other elements of the American Jobs Act in coming weeks, including money for building roads and schools, tax credits for businesses that hire veterans and the long-term unemployed, an extension of benefits for unemployed workers and an extension of a payroll tax holiday.

Democrats have been buoyed by polls that show providing dollars to pay teachers, police officers and firefighters is overwhelmingly popular, particularly as state and local budget cuts have resulted in teacher layoffs, increasing class sizes and eliminating of such school programs as art and music.

And they have pointed to monthly statistics that show the public sector has been losing jobs even as the private sector has been slowly adding employees.

But Republicans said the aid to states — $30 billion for education and $5 billion for public safety — would amount to a bailout for state and local government, paid for with new taxes on businesses best able to create private-sector jobs.

Twice before Congress has approved assistance for states, which were slashing their budgets because of declining tax revenue in the economic downturn. Economists say propping up state and local government budgets has slowed public-sector job losses, but GOP leaders counter that it has failed to spark a full economic recovery.

“What I’m saying is let’s put the government stimulus bills aside for a change and do something for the small-business men and women in this country who are begging for mercy from their own government in Washington,” Senate Minority Leader Mitch McConnell (R-Ky.) said.

McConnell accused Democrats of pushing the teacher and first-responder piece of the bill — along with the offsetting tax increase — to draw out Republican opposition. Even if the bill had passed the Senate, the GOP-led House would not have approved it.

To prove the point, McConnell forced the Thursday vote to repeal the withholding tax, which he said should have the support of senators in both parties.

Enacted in 2006, the withholding tax has been delayed repeatedly, mostly recently to the end of 2012, because of concerns that it would hurt companies that do business with the government.

But McConnell’s proposal to offset the tax’s repeal with $30 billion in budget cuts drew Democratic opposition and a veto threat from Obama. McConnell’s proposal drew a 57 to 43 vote, failing to reach the 60-vote threshold needed to end a filibuster and move ahead.