President Obama vowed this week to “talk about ways to improve the health care system,” a possible lifeline for Republicans who want to link Obamacare to a deal for government funding and increasing the nation’s debt ceiling.

Though the timing of such talks remains an issue — the White House says they should take place after funding and debt ceiling agreements, while Republicans want immediate discussions — there is growing traction on both ends of Pennsylvania Avenue for some changes to the president’s signature legislative achievement.

It’s no easy task.

Republicans will need to do more than just nibble at the margins of Obamacare if they want to appease staunch conservatives.

And the president won’t undermine his own blueprint as the administration works to sign up millions of Americans for health exchanges. Conservatives’ ultimate goal, scrapping the individual mandate, is going nowhere as long as Obama sits in the Oval Office.

However, there are a few alternatives that could achieve bipartisan support and possibly help broker a deal to end the shutdown.

Among the most realistic options on the cutting board: The threshold that employers must provide health insurance to those who work at least 30 hours a week.

By scrapping the 30-hour marker — and bringing the requirement more in line with the typical 40-hour work week — the White House would extend an olive branch to businesses who say they are using more part-time workers to avoid soaring health costs.

“That’s an attractive option,” one GOP House leadership aide said of eliminating the 30-hour threshold. “Look, we have to get something. There will be hell to pay [from the Republican base] if we don’t.”

The aide added that such a concession from Obama would “not be enough by itself” but could work as “part of a broader deal” on health care.

The administration has already delayed until 2015 the requirement that employers provide health insurance to their workers or pay a fine, and coupled with the litany of technological glitches since the online exchanges launched, the GOP sees an opening to make further changes to the law.

“How can we tax people for not buying a product from a website that doesn't work?" Speaker John Boehner, R-Ohio, said from the House floor on Wednesday.

Also, getting rid of the medical device tax has long been viewed as the lowest-hanging fruit of Obamacare’s less-popular provisions. The levy, which charges a 2.3 percent tax on sales of medical devices, faces opposition even from a handful of Democrats.

Such an approach isn’t without pitfalls, however.

“You take it off the books, how do you replace it?” asked Michael Tanner, a senior fellow at the Cato Institute, of the roughly $30 billion reduction in revenue if the device levy is eliminated. “Republicans certainly won’t agree to another tax. And nobody wants to increase the deficit.”

Analysts also point out that while eliminating the tax will increase profits for medical-device suppliers, it may not significantly impact the cost of health insurance.

For Republicans, the Obamacare negotiations could serve another task: putting red-state Democrats in tight spots.

Sen. Joe Manchin, D-W.Va., for example, has already voiced support for a one-year delay in the individual mandate. It’s a push that has been sidetracked by the constant focus on the debt ceiling and government shutdown battles, areas where Republicans see less political advantage than keeping the spotlight on Obamacare.

The more the White House and Republicans talk about Obamacare, the more likely Sens. Mark Pryor, D-Ark., and Mary Landrieu, D-La., — both seeking reelection — could be forced to hitch their wagons to provisions unpopular in their home states.

Republicans contend that the daily headlines about Obamacare glitches are only bolstering their case.

“How can the president not delay the law if it continues to go so poorly?” reasoned the House GOP leadership aide.

Also likely up for grabs during Obamacare negotiations are the insurance subsidies for members of Congress and their staffs.

A growing number of Republicans say the well-connected shouldn’t get “exemptions” from the law, and are making a push to end the practice even if they face opposition from their own staffs.