Hensarling said the duty to find healthcare reforms is "in many respects" more important than the supercommittee's more general goal of finding $1.5 trillion in deficit reduction over the next 10 years, as healthcare and other entitlement programs are major causes of the debt.

Hensarling noted that without an agreement in the supercommittee, the law still requires a minimum $1.2 trillion cut in the deficit over the next decade. But like other Republicans, he indicated some opposition to this sequestration of funds, which he said would fall too heavily on the Defense Department.

"It's not exactly across-the-board," he said. "Frankly, it disproportionately impacts national defense, in a way that even the secretary of Defense says will hollow out national defense."

Hensarling then seemed to imply that if there is no agreement within the supercommittee, Congress might be able to take this mandated $1.2 trillion in deficit reduction and tweak how those cuts are made.

"What I'm willing to do is be committed to ensuring that at least America gets that $1.2 trillion in deficit reduction," he said. "Now, Congress would have 13 months to do it in a smarter fashion."

The supercommittee now has less than 10 days to come to some agreement on how to reduce federal spending to hit the $1.5 trillion deficit-reduction target. Failing to reach a deal could trigger the sequestration process. However, Hensarling noted that in most cases, "cuts" to government spending would entail reductions in the rate of growth of federal programs to achieve a more sustainable pace.

"All we are talking about here is slowing the rate of growth," he said. "All of these programs, by and large, are going to continue to grow, but at a pace that would become more sustainable."

Republicans in recent weeks have again proposed some scaling-back of tax deductions in exchange for fundamental tax reform that leads to lower rates. On this, Hensarling said he could see a two-step process under which the supercommittee agrees to a certain amount of new tax revenue to be raised through tax reform, followed by work on the details of how to get there in various committees.