Republicans may be divided over the particulars of dueling budget plans, but there is at least one area of agreement among GOP lawmakers: the desire to repeal ObamaCare. 

And this year's budget legislation could give them the best chance yet to send repeal legislation to President Obama's desk. 

The party -- which for the first time in eight years controls both the House and Senate -- is using the budget process to inch closer toward a repeal of the controversial health care law, even as Obama hailed it a success on its five-year anniversary this week. 

The House on Wednesday night narrowly passed its version of a budget blueprint after Republican leaders agreed to tack on extra defense spending over the protest of conservative members who opposed busting the caps imposed by the 2011 Budget Control Act. The Senate, meanwhile, is on track to approve its plan by the end of week following a marathon voting session that starts Thursday. 

Leaders in both chambers have set a mid-April goal of resolving differences between the two approaches. 

Why does this matter for ObamaCare? 

Like many pieces of legislation before it, the proposal calls for undoing the health care law. What's significant this time is that it's contained in a budget resolution. 

While the resolution is not legally binding, it gives Senate leaders a procedural tool by which subsequent legislation -- so long as it impacts spending or revenue -- can pass the chamber on a simple-majority vote, as opposed to the usual 60-vote threshold. 

Known as budget "reconciliation," the tool is critical to GOP hopes of shelving ObamaCare, since Majority Leader Mitch McConnell's caucus of 54 is six votes short of filibuster-proof. (The budget itself is also subject to special rules and cannot be filibustered.) 

Five years after the president signed it into law, polls show that ObamaCare -- also known as the Affordable Care Act -- remains unpopular, with 42 percent of the public approving of it compared with 53 percent who disapprove, according to the Real Clear Politics average. 

The fact Obama likely would veto a rollback of his signature legislative achievement isn't stopping Republicans from fulfilling what they describe as a campaign promise. 

"This new class, we were elected on ObamaCare, to repeal and replace it," Sen. Cory Gardner, a freshman Republican from Colorado, told Fox News' Neil Cavuto on "Your World" earlier this week. "I think the Republicans not only will have a plan, but something the president will accept, because it's something that we have to do, and that's important." 

Democrats nevertheless continue to trumpet individual parts of the overhaul, citing benefits like the elimination of preexisting conditions as grounds for denial of coverage and the expansion of Medicaid to cover more Americans. For his part, Obama acknowledges there may be room for improvement, but says Republicans should work with Democrats to address problems rather than aim for a wholesale repeal. 

"Every public health policy has some tradeoffs, especially when it affects one-sixth of the American economy and applies to the very personal needs of every individual American," the president told an audience Wednesday. "We also know beyond a shred of the doubt the policy has worked. Coverage is up, cost growth is at a historic low, deficits have been slashed, lives have been saved. So if anybody wants to join us in the spirit of people putting aside differences and come here today and make the law work better, come on board." 

But even if Obama were to veto their efforts, Republicans -- who have voted more than 60 times to repeal or undermine the law -- are champing at the bit to finally send legislation to his desk. 

"By passing a balanced budget that's about the future, we can leave ObamaCare's higher costs and broken promises where they belong -- in the past," McConnell said Monday. 

Details of a possible GOP replacement for ObamaCare remain elusive for the time being. There is no shortage of Republican-sponsored health-care bills percolating on the Hill but it isn't clear if the party will unite around one comprehensive magic bullet. 

Indeed, given the likelihood of a veto many ObamaCare critics say the best shot they'll have at undermining the law could be in responding to a forthcoming Supreme Court ruling in King v. Burwell. In the event the high court rules against the administration, some 7.5 million Americans enrolled in federally run health-care exchanges could lose their subsidies as a result, according to a February estimate by the Kaiser Family Foundation. 

"I look forward to that opportunity," said Sen. Bill Cassidy, a freshman Republican from Louisiana who is also a medical doctor. "If it arises, we shall be ready." 

Capitol Attitude is a weekly column written by members of the Fox News Capitol Hill team. Their articles take you inside the halls of Congress, and cover the spectrum of policy issues being introduced, debated and voted on there.