Obama had repeatedly promised that Americans who liked their health insurance could keep it when the law took effect on Oct. 1, but many people have received cancellation notices because their insurance plans do not meet the standards set out in the Obamacare law.
"As I indicated earlier, I completely get how upsetting this can be for American after assurances they heard from me," said Obama. "To those Americans, I hear you loud and clear and today I'm offering an idea that could help do it."
As of Wednesday, just over 106,000 people nationally signed up for Obamacare health insurance plans, according to the federal figures, only a small portion of the millions who had been expected to enroll. And only 26,000 of those individuals went through the HealthCare.gov site.
(Read more: Obamacare rollout numbers worse than feared)
The Obama administration had warned that enrollment would be very low in October due to website glitches.
"The rollout has been rough so far and everyone understands I'm not happy about the fact that the rollout has been wrought with a whole range of problems I've been deeply concerned with," said Obama.
Obama said he was not informed directly that the Obamacare website would not work the way it was supposed to.
Obama's top technology officials testified at a House oversight hearing Wednesday.
"We have much work still to do, but are making progress at a growing rate," Todd Park, the chief technology officer at the White House, told the Republican-led House Oversight Committee.
(Read more: Obamacare Web model: Critics bullied me)
—By CNBC's JeeYeon Park (Follow JeeYeon on Twitter:
@JeeYeonParkCNBC). Reuters contributed to this article.