"We're more than happy to negotiate with the president about how you distribute the cuts … but the idea that there's going to be any more revenue here, that's just not going to occur. These cuts are in law, they're going to happen," said Cole on MSNBC's "Morning Joe."
"We'd like to work to redistribute them, but again, they're going to take place and they need to take place," he added.
"We'll probably do that within a month in the continuing resolution ourselves, if the president doesn't come to the table," he said.
President Obama has called for a package that would replace the sequester with a combination of spending cuts and new tax revenues. But GOP lawmakers say they addressed new revenues in January's "fiscal cliff" deal and want any sequester replacement to focus on cuts and entitlement reform.
Cole said that the "real money" for deficit reduction is in entitlement programs.
"We're going to have to reform those programs or we're going to lose them. And I don't want to lose Social Security or Medicare or Medicaid, but they've got to change dramatically. We tried to do that in the [Rep. Paul] Ryan [(R-Wis.)] budget last year. I think you'll see us even more aggressive this year, so that's where we ought to go, but again, we're not going to lose these cuts. They're in law," he added.
In order to reform the non-discretionary programs, Cole called for linking and gradually raising the eligibility age of Medicare and Social Security and moving to chained CPI right away.
"The real drivers, again, are long term, and you begin to gradually change that and you change the fiscal outlook of the country a lot quicker than you do with sequester," Cole said.
Both parties have expressed concern about the effect the across-the-board spending cuts will have on the Pentagon, with Defense Secretary Leon Panetta warning they could hamper U.S. military capabilities.
Cole said he agreed with former Joint Chiefs Chairman Adm. Mike Mullen's assessment that the "biggest national security threat to the country in the long term is the deficit."
"That's going to affect the military, not just today or tomorrow, but indefinitely. Having said that, I would prefer not to do this. We've taken half a trillion dollars out of defense already," Cole said.