As thousands of consumers receive health insurance cancellation notices, President Obama is continuing to fine tune his promise that Americans who like their health plans would be able to keep them under his signature health care law.

The president told about 200 of his campaign supporters and health care activists Monday that the administration had promised Americans they could keep their current coverage -- as long as their plans hadn’t changed since ObamaCare was signed into law.

"If you have or had one of these plans before the Affordable Care Act came into law and you really liked that plan, what we said was you could keep it if it hasn't changed since the law's passed," Obama said. "So we wrote into the Affordable Care Act you are grandfathered in on that plan. But if the insurance company changes it, then what we're saying is they have got to change it to a higher standard. They've got to make it better."

Obama said his health care law is making the insurance market better for everyone, even though millions of Americans who buy individual plans have been receiving cancellation notices. He said the notices "can be scary for people," but added they were often getting "a very bad deal."

"The bottom line is -- is that we are making the insurance market better for everybody and that's right thing to do," he said.

The president also acknowledged that technical problems with the health care website to shop for insurance are making it tougher for people to find other plans. He said he's "not happy about it," but promised, "We're going to smooth this thing out."

At least 3.5 million Americans have been issued cancellations, but the exact number is unclear, The Associated Press reported. In a speech at Boston's Faneuil Hall this past week, Obama said the problem is limited to fewer than 5 percent of Americans "who've got cut-rate plans that don't offer real financial protection in the event of a serious illness or an accident."

In the Republican-controlled House, officials say a vote is likely as early as next week on a bill to let insurers continue selling any individual policies that were in effect on Jan. 1, 2013, even if they provide coverage deemed insufficient under Obamacare.

The measure, drafted by Rep. Fred Upton, R-Mich., and backed by roughly 100 fellow Republicans, would remain in effect throughout 2014, after which the issue would presumably be reviewed.

"Despite the president's repeated promise of 'if you like your plan, you can keep it,' many Americans are now learning the sad reality that their current plan will no longer exist beginning on January 1," Upton, chairman of the Energy and Commerce Committee, said in a statement when he announced his legislation last week.

"This legislation is about providing folks the peace of mind that they will be allowed to keep their current coverage if they so choose."

While Upton's legislation permits insurers to sell existing coverage plans that would otherwise be banned, Sen. Mary Landrieu, D-La., introduced legislation during the day to go one step further by requiring it.

Aides told The Associated Press that under her measure, insurance companies would be obliged to continue offering existing paying customers continued coverage under any plans in effect at the end of 2013. No new consumers could enroll.

"A promise was made that if you like your health plan, you can keep it - and I will do everything I can to see that the promise is kept," said Landrieu, who is gearing up for what is expected to be a challenging re-election campaign next year.

The White House declined comment on the two bills.

Both measures are designed to cut through a complicated system in which some plans for individuals that fall short of coverage standards are allowed to continue under Obamacare, while others are not.

The Associated Press contributed to this report.