Paul was referring to a bill from Senate Majority Leader Harry ReidHarry Mason ReidDems search for winning playbook Dems face hard choice for State of the Union response The Memo: Immigration battle tests activists’ muscle MORE (D-Nev.), S. 1323, which is expected to come up for a vote on Thursday. Reid filed cloture on his bill Tuesday, after acknowledging that the upper chamber needs to make a priority of resolving the debt-ceiling crisis.

But Paul argued that Reid's bill is the wrong answer.

"Unfortunately I don't think the Democrats are serious about this," he said. "They've produced a resolution that says they can raise taxes, which is a non-starter; it's a horrible prescription for an economy in the middle of a recession, and isn't going anywhere."

Like several other Republicans today, Paul argued that the U.S. is deeply in debt, and said the only solution is to cut spending. He argued specifically that while the U.S. has been able to borrow billions of dollars at historically low interest rates, those rates could jump and add billions more to U.S. interest payments.

"If interest rates go back to the historic average, we will be swamped in debt," he said, adding that interest payments could top $5 trillion over the next 10 years. "This is what looms."