But it remains uncertain what the Senate will do.
Last week, Senate Democrats prevented similar legislation from reaching the floor; they objected to covering the cost of the repeal by directing the executive branch to rescind funds and make further cuts to discretionary spending in 2012. The White House threatened to veto that measure because of those pay-fors.
But the Republican-pushed House bill passed on Thursday would cover the cost of undoing the contractors' tax with a fix to the 2010 health care law that would tighten Medicaid eligibility requirements, saving $14 billion. That fix was passed earlier on Thursday, 262-157.
The Obama administration has backed the House bill. But how a final version will be worked out with the Senate remains ambiguous. The Office of Management and Budget has also said the administration would be willing to work with Congress to identify "acceptable offsets" for the repeal's budgetary costs.
Senate Majority Leader Harry Reid, D-Nev., said last week he would take up the House bill, though he said he wanted to pay for it with elimination of a corporate-jet subsidy and tax breaks for multinational corporations, an approach Republicans oppose.
Some Democrats are reluctant to use the Medicaid savings to offset the cost of eliminating the withholding tax, preferring to save the offset for other legislation. Whatever the approach to paying for it, Reid said he was confident the withholding tax will be repealed. "I don't think people are too far apart on this," Reid spokesman Adam Jentleson said.
The contracting rule was originally devised in 2006 as a means of combating tax avoidance, but has never been implemented and has drawn the ire of Republicans and Democrats alike as a burdensome requirement on potential job creators.
Implementation had been delayed earlier this year until the end of 2012. But President Obama proposed in his jobs package this fall that the regulation be delayed until 2014.
The Medicaid-related bill is a rare glimmer of bipartisanship when it comes to the 2010 health care law. The White House and some congressional Democrats, although not all, want to see the "fix" passed to alleviate a larger budget crunch on the already-strapped Medicaid program.
The bill would undo a piece of the 2010 health care overhaul that would have made more than a million seniors newly eligible for Medicaid, because Social Security income would no longer factor into eligibility.
The president included the $14 billion cost-saver in his latest deficit-reduction package, and the White House did not threaten to veto the House bill passed on Thursday sponsored by Rep. Diane Black, R-Tenn.
But House Democrats largely voted against that bill because they would rather use the Medicaid funds to cover the cost of other health-related bills, like the looming Medicare doctors' pay fix that Congress must deal with by the end of the year.
"It certainly has been raised in my caucus," Rep. Allyson Schwartz, D-Pa., said in an interview.