"A number of people are working on it, and we've come up with the various provisions, and now hopefully we're going to put together a Republican package" on health care, McCain told Reuters outside the Senate last week.
Several bills have already been introduced by Republicans in the House and Senate but no single plan has yet emerged.
Some start with the repeal of the Patient Protection and Affordable Care Act—a move that would almost certainly be vetoed by Obama if it passed both chambers, which is unlikely as long as Democrats hold the Senate.
Some bills propose new tax credits or deductions to help people pay for health insurance.
(Read more: Obamacare has 'not been great': Boots chairman)
The law, commonly called Obamacare, passed Congress in 2010 as the most sweeping U.S. social legislation in 50 years and survived a legal challenge by opponents in the U.S. Supreme Court in 2012.
It requires most Americans to buy insurance, offers subsidies to help low-income people receive coverage and sets minimum standards for coverage. It aims to dramatically reduce the number of Americans who lack health insurance policies.
Instead of starting with a total repeal, Republican Senator Ron Johnson of Wisconsin has suggested transition legislation that might initially eliminate some provisions such as mandatory coverage of maternity care and move people with pre-existing conditions into high-risk insurance pools.
Some tea party-backed House conservatives also are urging action, arguing that it may no longer be enough to simply denounce Obamacare as lawmakers start campaigning for congressional elections in November.
"What's our alternative to this terrible thing called Obamacare?" asked Representative Jim Jordan of Ohio at a recent lunch meeting of House Republican conservatives.
Another conservative, Representative Raul Labrador of Idaho, said that if Republicans want to win in 2014, they should start "letting the American people know what we are for."
Boehner, who has presided over dozens of House votes to limit or curtail Obamacare, said that at their annual retreat Jan. 29-31, House Republicans would discuss a plan to make health care insurance more accessible and affordable.
What do Republicans want?
Republicans have opposed the law for years. They say Obamacare relies too heavily on mandates and results in too much government interference in the marketplace.
They point to the rocky rollout of the Obamacare website last October as evidence of flaws in the law. But analysts wonder whether they can unify around an alternative.
"The Republicans historically had a lot of health care bills ... The Republicans never coalesced around a single bill, and that was the political weakness of the Republicans," said Bob Moffit, a senior fellow at the conservative Heritage Foundation who was a top health official under President Reagan.
(Read more: No Medicaid expansion? There's an Obamacare tax hit)
The administration says Obamacare is settled law now.
Vice President Joe Biden on Thursday touted popular provisions such as prohibiting insurers from rejecting people with pre-existing conditions. "We will not go back. America has turned the page."
Republicans are not averse to cherry-picking some of the more popular bits of Obamacare. Two separate House Republican proposals would address the needs of people with pre-existing conditions through state-run "high-risk" insurance pools.