"Why would you pass a debt limit ... without doing something about the deficit?" he asked rhetorically on "CNN Newsroom" on Monday.
He pointed to the 2011 debate over the debt limit that reduced the deficit from $1 trillion to $700 billion.
"We're saying we should do the same thing now," Cole added.
Additionally, some Republicans see the debt limit fight as an opportunity to not only shrink the size of government but also to pay off interest.
Republican Sen. Tom Coburn of Oklahoma said the United States could meet its obligations without lifting the debt ceiling.
"I would dispel the rumor that is going around that you hear on every newscast that if we don't raise the debt ceiling we will default on our debt," Coburn said Monday on CBS' "This Morning." "We won't. We will continue to pay our interest. We'll continue to redeem bonds and we'll issue new bonds to replace those."
What Coburn doesn't mention is that the United States would have to choose to pay its interest over the billions of dollars of other obligations -- Medicare, Social Security, etc. -- for which the federal government is responsible.
Lew, however, said that's a "dangerous" and "reckless" notion.
"[T]he reality is, there are no good choices if we run out of borrowing capacity and we run out of cash," he said.
Most economists tend to agree with Lew's assessment.
Moody's economist Mark Zandi told the Joint Economic Committee recently that the economy "would quickly fall into another severe recession" if the debt ceiling is not lifted in adequate time.
Which side is right? Before default is reached, it's hard to know.
But a recent study published in Psychology Science found that people on the far end of the political spectrum believe that their positions are "correct."
"Given the stalemate in Washington, understanding why people become so entrenched in their views --- even when there is not an objectively correct answer --- is more important than ever," Kaitlin Toner, a postdoctoral fellow at Vanderbilt University, said, describing why she conducted the study.
Wall Street could be the biggest indicator of who is right in this fight. Investors nervous over the political impasse in Washington pushed all three major U.S. stock indexes lower on Monday. The Dow Jones Industrial Average alone lost more than 130 points, or nearly 1 percent of its value.