"This fix won't solve every problem for every person," Obama said, "but it's going to help a lot of people."
Republicans, who have fought against the health-reform law since its passage in 2010, were unimpressed with Obama's proposal.
House Speaker John Boehner said it was time to "scrap this law once and for all."
"You can't fix this government-run health care plan called Obamacare. It's just not fixable," he added, according to the Associated Press.
The proposed change to the health law comes as the Obama administration works to regain credibility after the disastrous launch of the new online federal health marketplace -- or exchange -- where consumers may buy health insurance, in many cases with the help of federal tax credits.
"That's on me. We fumbled the roll-out on this health-care law," Obama said Thursday, adding that he's confident that next year people will look back and find that the new law is working well.
In addition to fixing the HealthCare.gov website, Obama said his administration would be taking steps to make the process of purchasing health insurance simpler by streamlining the application process, giving people more assistance and communicating people's health-plan options in a clear way.
In response to Obama's remarks, America's Health Insurance Plans, an insurance trade association, issued a statement saying that consumers may still face higher premiums.
"Changing the rules after health plans have already met the requirements of the [health-reform] law could destabilize the market and result in higher premiums for consumers," AHIP president and CEO Karen Ignagni said in a news release. "Premiums have already been set for next year based on an assumption of when consumers will be transitioning to the new marketplace.
"If now fewer younger and healthier people choose to purchase coverage in the exchange, premiums will increase and there will be fewer choices for consumers," she added. "Additional steps must be taken to stabilize the marketplace and mitigate the adverse impact on consumers."
Ignagni said that the "only reason consumers are getting notices about their current coverage changing is because the ACA [Affordable Care Act] requires all policies to cover a broad range of benefits that go beyond what many people choose to purchase today."