The across-the-board spending cuts known as the sequester have been criticized by both parties as an unsophisticated way to reduce spending. But Republicans and Democrats have so far found no other way to trim the budget deficit — Republicans would prefer more cuts, but Democrats have insisted on new taxes.

WATE also spoke with Sen. Bob CorkerRobert (Bob) Phillips CorkerSanford: GOP lawmakers 'running for cover' over fear of Trump tweets Trump’s trusted diplomat faces daunting task with North Korea George Will says Trump doesn’t inspire ‘cult’ in GOP: ‘This is fear’ MORE (R-Tenn.), who said the key is to phase in spending cuts over the course of a decade.

"You can do these things in an artful, graceful way so that you almost don't notice that it's occurring," he said. "But you have to start turning the ship. And everybody acts like something draconian is going to happen the next day. That's not the case. These are tweaks."

Corker said he was optimistic that some deal on spending can get done. "There's beginning to be, in my opinion, the right environment for these solutions," he said.

When Congress returns next week, discussions are likely to start on how to get around the debt ceiling problem in mid-May. On May 19, the current freeze on the debt ceiling will be lifted.

An increase in the ceiling may not be needed immediately, as some have said Treasury might be able to manage the debt for several months. Still, the House is expected to pass a bill in May that would ensure the government continues to pay interest it owes on the debt in the event the government does run out of authority to borrow money.

Despite Corker's optimism, Democrats in the House and Senate have said they oppose the payment prioritization bill, and there have been few public signs that the two parties are close to a debt ceiling agreement.