Updated 11/12/2013 at 2:19pm ET

(CNN) – President Barack Obama has already apologized to Americans who lost their health plans under the Affordable Care Act. Now Bill Clinton is calling on his fellow Democrat to find a way to uphold an earlier vow that those who like their insurance plans can keep them.

"I personally believe, even if it takes a change in the law, the president should honor the commitment the federal government made to those people and let them keep what they've got," Clinton said during an interview with the website OZY.com.
 
Clinton did, however, spend much of the interview defending the 2010 health care overhaul.

"The big lesson is that we're better off with this law than without it," he said.

"The enrollment period didn't come off well because the national website wasn't ready," Clinton noted. "But this happened once before. It happened when President Bush put in the Medicare drug program for seniors, which was not as complicated, but had exactly the same problem with the roll out. It was a disaster. There were people that lost their prescriptions for their existing medicine. And they fixed it."

Obama agrees with Clinton's assessment, White House Press Secretary Jay Carney said Tuesday.

The President "has pledged to task his team" to find solutions for people who have lost their plans despite assurances made before the law's passage, Carney told reporters during the White House press briefing.

Carney also said – as administration officials have previously noted – that the initial Obamacare enrollment numbers will be released by the end of the week, and “will be lower than we hoped and we anticipated.”

Also wading into the controversy, former GOP vice presidential nominee Sarah Palin had some much tougher words for Obama on Tuesday.

In an interview with CNN's Jake Tapper, Palin recalled statements Clinton made during the 2008 Democratic primaries, when the former president was stumping for his wife.

"I'll never forget Bill Clinton saying about Barack Obama and his story, his agenda, that it was the biggest fairy tale he'd ever seen," Palin said.

"And he was right, because Barack Obama was not qualified, he was not prepared, and the manifestation of that today is the problem that we see left and right in our economy," she added.

The full interview on "The Lead with Jake Tapper" will air at 4 p.m. ET on Tuesday.

For his part, Speaker John Boehner said Tuesday that Clinton’s comments “signify a growing recognition that Americans were misled when they were promised that they could keep their coverage under President Obama’s health care law."

The House of Representatives is slated to vote later this week on a GOP-backed measure that would enact changes to Obamacare allowing insurance companies to continue offering individual plans that are currently being canceled because they don't meet the Affordable Care Act's standards for coverage.

While congressional Democrats are largely standing by the law, a number of them are trying to distance themselves from problems tied to the rollout. Among other things, some Democrats are pushing for an extension of the enrollment period or a delay in the penalty for not obtaining coverage.

Sen. Dick Durbin of Illinois, the number two Democrat in the Senate, said Tuesday it was a mistake for Obama to have not fully explained his comments on individuals being able to keep their plans.

"A couple more sentences added would clarify it," Durbin told CNN. "The President apologized. He said very clearly he was sorry if he misled people."

When Obama apologized last week to people who were losing their health coverage "based on assurances they got from me," he added that he was asking his advisers to look at ways in which the law could be altered.

"I've assigned my team to see what we can do to close some of the holes and gaps in the law, because, you know, my intention is to lift up and make sure the insurance that people buy is effective – that it's actually going to deliver what they think they're purchasing," Obama said.

Officials later said those changes could include administrative fixes to the law.

–CNN's Bryan Koenig and Sherisse Pham contributed to this report.