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The demand headlines a conservative wish list that the party is set to put forward in negotiations over the federal borrowing limit.
President Barack Obama hit back by accusing Republicans of "crazy" scaremongering about the health law.
The US is set to hit the debt ceiling on 17 October, leaving the government short of money to pay its bills.
It is one of two fiscal deadlines that have again left Congress at a stalemate.
In addition to the debt ceiling, the federal government will shut down most of its operations if a budget bill to maintain its funding is not passed by 1 October.
"We have no interest in seeing a government shutdown," Republican House of Representatives Speaker John Boehner told reporters on Thursday. "But we've got to address the spending problem."
'50 favourite flavours'
House Republicans have already inserted a provision that would strip funding from the healthcare law into their version of a spending bill to avert a government shutdown, which is due to happen next Tuesday.
As the hours tick by to a shutdown, the Republican leadership is sounding a little desperate, and the Democrats are enjoying watching them squirm ”
But the Senate will strike out that proposal from its version of the bill. Mr Boehner did not say on Thursday whether his party would accept such a revision.
Analysts say Republicans have seemed in recent days to be backing away from a showdown over the shutdown, in case it backfires politically.
On Thursday, the party brought its fight against President Barack Obama's healthcare law to the negotiations over the debt ceiling.
The White House and its Democratic allies have repudiated legislative attempts to target the Affordable Care Act, commonly known as Obamacare.
They point out it has been the law of the land since 2010 and withstood a US Supreme Court challenge.
Republican House Majority Leader Eric Cantor, the number-two Republican in the chamber, told Thursday's news conference that the debt ceiling bill would be put forward this week.
The National Review magazine reports that an outline of the Republican proposal includes changes to tax laws, cuts to civil service pensions, an overhaul of financial regulations and means-testing for federal benefits.
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What the papers say
The Washington Post's Wonkblog says: "John Boehner isn't even trying to pretend his House of Representatives is a sane place anymore.... This looks like an Onion parody of what the House's debt-ceiling demands might be. It's a wonder it's not written in comic sans."
A New York Times editorial says: "The budget crisis manufactured by Congressional Republicans will never succeed at halting health care reform, but it has already caused long-lasting damage to the economy."
The Wall Street Journal writes: "The emergence of the health law as Republicans' top target reflects the resurgence of tea-party-fueled activism."
The House Republican bill will also reportedly demand approval for the Keystone XL pipeline, increased offshore oil drilling and a rollback of environmental regulations.
But Patty Murray, the Democratic chairwoman of the Senate budget committee, told the New York Times: "This is not the time to throw in your 50 favourite flavours."
Mr Obama made an impassioned defence of his signature healthcare overhaul during a speech on Thursday at a college in the US state of Maryland.
With just five days to go before Americans can begin signing up for medical coverage under the law, the president poked fun at his Republican opponents, saying: "The closer we get, the more desperate they get."
