* "Obamacare" fades in Republican rhetoric over debt limit
* Some Republicans welcome return to deficits, spending
By David Lawder and Caren Bohan
WASHINGTON, Oct 9 (Reuters) - In a shift some Republicans hope will strengthen their hand in the fight over the U.S. debt limit, some of the party's leaders in the House of Representatives are playing down demands to weaken "Obamacare" and focusing instead on calls to rein in deficits.
The emphasis on tackling long-term debt and deficits was evident in opinion pieces published on Wednesday by two of the most prominent House Republicans, Majority Leader Eric Cantor and Paul Ryan, the chairman of the House Budget Committee and a former vice presidential candidate.
Ryan's piece in the Wall Street Journal failed to mention Obamacare, the word Republicans coined to describe President Barack Obama's signature healthcare reform law, while Cantor's article in the Washington Post made only a passing mention as he criticized Obama for refusing to negotiate over the debt limit.
Ryan proposed that Republicans and Democrats work out a plan to curb costly entitlement programs and overhaul the tax code as part of negotiations to end the U.S. government shutdown and raise the country's borrowing limit.
"This isn't a grand bargain," Ryan wrote, but would be a way to "get a down payment on the debt and boost the economy."
The effort to scuttle Obama's healthcare law was at the center of the Republican strategy for weeks during the run-up to the shutdown that started on Oct. 1.
In exchange for the funding needed to keep government agencies open, Republicans first sought to deny money for the health-insurance reforms, saying later they would settle for a delay of the program.
Neither is acceptable to congressional Democrats or Obama.
Responding to Ryan's opinion piece, Amanda Carpenter, speechwriter and senior communications adviser to anti-Obamacare leader Texas Senator Ted Cruz, tweeted: "There is one big word missing from this op-ed. It's start with an O and ends with BAMACARE."
Jenny Beth Martin, National Coordinator for Tea Party Patriots, one of several national groups that campaigned to "defund Obamacare," also criticized Ryan for the failure to mention the health care law.
"We must remember the reason we are fighting and remain united in our opposition to Obamacare," Martin said in a statement.
But some in the party insist it is time to move on to more fertile ground, and say the budget is the obvious place to go for debt-limit talks.
Representative Lee Terry, an Iowa Republican, saw it as a "natural" migration as the debt-limit deadline approaches. "It was a good try," Terry said of the effort to undermine Obamacare.
While there may be an opportunity to push for small changes, such as a repeal of a medical-device tax used to help fund the healthcare program, he added, "I think we've run the course on Obamacare."
The linking of Obamacare to the budget fight was a strategy championed by Cruz and pushed by a vocal faction of House of Representatives conservatives, led by members backed by the small-government Tea Party movement.
Ryan, Cantor and Boehner embraced the strategy only reluctantly from the outset.
Some conservatives are now seeking to rebrand the campaign against Obamacare as part of a broader attack on entitlement spending that should be cut in any talks over the debt limit.
House Speaker John Boehner continues to mention it in his statements to the press, including on Wednesday. "Instead of making it easier for people to get health insurance, it is going to be lot tougher," Boehner he said on the House floor. "What a train wreck."
Scott Garrett, a senior Republican on the House Budget Committee, told CNN the Affordable Care Act, as Obamacare is formally known, was "a brand-new entitlement to the tune off $2 trillion. So all those things need to be on the table."
U.S. Treasury Secretary Jack Lew has warned U.S. lawmakers that unless Congress raises the debt ceiling by Oct. 17, the country could be at risk of a first-ever default.