A blunt critique from Bill Clinton on President Obama's handling of the rocky ObamaCare rollout is prompting the White House to pledge another set of health law fixes -- though in doing so, it could inadvertently build the case for those calling for a delay in the law's implementation.

Aside from scrambling to fix the broken HealthCare.gov website, the administration is now trying to deal with the millions of Americans who have received cancellation notices. On Tuesday, White House Press Secretary Jay Carney said the president's team is trying to figure out a way to offer relief to some of those policyholders.

The statement comes after former President Clinton, in an interview with the site Ozy.com, said Obama should live up to his promise to Americans that if they like their health plans, they can keep them.

"So I personally believe, even if it takes a change to the law, the president should honor the commitment the federal government made to those people and let them keep what they got," Clinton said.

Asked Tuesday if Obama agreed, Carney said: "The answer's yes."

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Carney referred to Obama's comments to NBC News last week, when the president apologized for the cancellation notices and said the administration was looking at a "range of options" to address the issue.

But Carney spoke in additional detail on Tuesday, saying the president's team is specifically trying to help those Americans who have been forced off their plans into more expensive coverage which they might struggle to afford.

"The president has tasked his team with looking at a range of options, as he said, to make sure that nobody is put in a position where their plans have been canceled and they can't afford a better plan, even though they'd like to have a better plan," Carney said.

It's unclear, though, how the administration could go about doing that. Insurance companies are the ones canceling policies -- due to new requirements under the health care overhaul -- and would likely have to be involved in any effort to restore coverage.

Robert Zirkelbach, spokesman with America's Health Insurance Plans, told FoxNews.com that all the myriad plans to reverse or stop the cancellation notices raise questions about how that would be achieved. He said in many cases, changes to existing policies have already been filed with state regulators and consumers have been notified.

"That's a lot to go back and unwind," he said, noting that any modified health plans would have to be reviewed and approved by state regulators.

"That typical process takes many months," he said.

The insurance industry generally does not support any delay in the implementation of the ObamaCare insurance mandates, out of concern that a delay would deprive them of much-needed customers at a time when they're dealing with the cost of additional coverage requirements.

But any effort to overhaul the cancellations might require an extended timeframe.

Separately, House Republicans, as well as some Senate Democrats, are planning legislation that would allow people losing their current coverage to keep their plans.

A Senate bill introduced by Louisiana Democratic Sen. Mary Landrieu that would ensure Americans could keep their current health insurance plans was co-sponsored Tuesday by longtime Sen. Dianne Feinstein, D-Calif.

In his interview Tuesday, Clinton defended the health care law as a whole, but explained how the broken promise on health coverage can hurt young people. He relayed the story of a young man who said his individual market plan was canceled and replaced with one whose premiums were twice as high. Though his deductibles and co-pays were lower, that savings is only realized if he gets sick, Clinton explained.

Obama, in explaining the cancellation notices, has clarified that under ObamaCare, policies could be canceled if they had been altered in any way since the passage of the law.

That nuance was not included in the president's initial explanations.

The administration argues that while some are losing their current coverage, those plans will be replaced by better-quality insurance. The flip side is that they could be more expensive.