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Sen. Lamar Alexander is introducing a bill that would push the government to release weekly updates on Obamacare figures.
"All we want to know in the Internet age is how many people tried to get on, how many were successful, what's their zip code, what was their level of insurance," said the Tennessee Republican.
White House spokesman Jay Carney said Wednesday the Department of Health and Human Services will begin conducting daily briefings on Thursday about progress toward fixing problems with the Healthcare.gov website.
Alexander has been a vocal critic of the Affordable Care Act, but conservative Republicans were not impressed with his role, or lack of one, in attempts to defund the law during the latest fiscal showdown.
Alexander is up for reelection next year, and has already been the target of attack ads.
"I admire the passion of all those who wanted to defund Obamacare, but I don't admire their tactics," said Alexander.
Conservative, mostly tea party backed Republicans refused to pass a government funding bill that did not include defunding or delaying Obamacare, which lead to the 16-day government shutdown this month.
"I want us to be the take-over-the-government party, not the shut-down-the-government party. That means we need to elect six more Republican Senators, the president in a couple of years. Then we could really change Obamacare," said Alexander.
There are 760,000 adults and more than 100,000 children without health insurance in Tennessee.
Asked if a functioning healthcare.gov – the federal health exchange website – could help those people get coverage, Alexander replied, "I think it could." But he added that many of the more than 800,000 uninsured in his state would be eligible for Tennessee's Medicaid program.
"At least Tennesseans need to know what the cost will be," said Alexander.
The federal law states that uninsured Americans need to sign up and purchase insurance by December 15, in order to be insured by January 1.
"And you may be fined by the IRS if you don't, for trying to enroll on a website that doesn't work. So we need to know if that's the case. Already one Democratic Senator has said it's going so badly perhaps they should delay implementation of the individual mandate," said Alexander, adding he himself would suggest delaying it for at least one year.
"That would also give us a chance to have a referendum in 2014 on whether we ought to make dramatic changes in the health care law, which I think we should," said Alexander.
Health and Human Services Secretary Kathleen Sebelius told CNN's Sanjay Gupta that President Barack Obama did not know of the problems plaguing the healthcare.gov website ahead of its October 1 launch date.
"They should have. I mean, the president himself seems embarrassed by it. And if he's not going to resign over this mess, why, he ought to decide who should," said Alexander, who is not joining calls from some of his colleagues that Sebelius resign.
Alexander said the "terrible results" of the website are "a grave embarrassment. And that's why I think my bill's pretty important."
Alexander said disclosing details like how many people tried to get health insurance, how many actually got it, and what level of insurance they bought "would actually build confidence in the system if it was working."
"And if it's not working, it would give us a chance to insist that it be fixed," said Alexander.
CNN's Jessica Metzger contributed to this report.