Both Sen. Dianne Feinstein, D-Calif., and Rep. Mike Rogers, R-Mich., said on CBS' "Face the Nation" that the HealthCare.gov website should go offline until it is fully functional.
The website has been plagued by problems since its Oct. 1 launch. Earlier this month, President Obama appointed former CEO Jeffrey Zients, one of his economic advisors, to lead a "tech surge" intended to fix the site by the end of November.
"It's pretty clear, I think, to those of us who have been watching this rollout, that the technological base was not sufficient, and that the website didn't function," said Feinstein, who added that she told White House Chief of Staff Denis McDonough that she thought the administration should take the site down "until it was right."
Still, Feinstein said that technological problems are not indicative of a larger problem with the law.
"The fact is there's a problem with the start up. I won't go so far right now as to say the plan isn't going to work," she said. "This is a very large, major priority and if it can get up and running it can be, I think, a very positive thing."
Feinstein blamed problems in part on "so many destroyers" that don't want to see the law work in the public, the House of Representatives and the private healthcare sector.
Rogers, who questioned Health and Human Services Secretary Kathleen Sebelius during a hearing last week, has taken issue with concerns about the safety of Americans' healthcare information.
"What was really shocking to me is even by their own words, they admitted that there was a high degree of risk when they offered the website to go public, they never told anybody about that. They said that they think the risk was acceptable. But their information wasn't at risk, American people's information was at risk," he said.
Rogers likened the administration's attempts to fix the website while it is running to "trying to change a tire on a car going 70 miles an hour down the expressway."
"They need to take the site down, stabilize it, meaning they can't continue to add code every week and then they need to stress test the system," he said.
Host Bob Schieffer also asked Feinstein about the president's claim that people would be able to keep their health insurance if they liked it, which has shown to be untrue.
"It is really very unclear right now exactly what the situation is. And yes, that's a problem," Feinstein said.
The White House has stood by the pledge. "If you had one of these substandard plans before the Affordable Care Act became law, and you liked that plan, you were able to keep it," the president during a speech in Boston last week.