While he would not provide specifics of the deal being negotiated, the six-term lawmaker characterized it this way:
"It's likely to be a pretty small deal, something that trades some relatively modest savings and perhaps some small entitlement reforms and maybe a little dab of revenue somehow."
Such a deal, Cole said, would "loosen up the sequester" that requires annual across-the-board spending cuts of around $100 billion a year. Those cuts would hit the Pentagon and an array of other agencies that oversee everything from environmental programs to medical research and national parks.
(Read more: Obamacare may need a taxpayer bailout: Ex-HHS head)
User fees, spectrum sales
The deal could reduce the sequester cuts by around $100 billion, but would be spread over multiple years, Cole said.
"But that's still pretty fluid," he added.
The "dab of revenue," Cole said, might come in the form of non-tax user fees or other steps, such as the government sale of some communications spectrum.
Republicans have firmly opposed tax revenue increases as part of any deficit-reduction efforts. Instead, they have pressed for reductions in Medicare and Medicaid health care spending and the Social Security retirement and disability program as a way of replacing the automatic cuts that began in March.
Democratic leaders have refused to consider trimming those programs unless they get some tax increases in return.
(Read more: US deficit falls 24% in October due to shutdown)
Senate Budget Committee head Patty Murray, the Democrat who leads the special budget negotiating committee along with House Budget Committee Chairman Paul Ryan, told leaders of Hispanic groups on Wednesday that she still wants to raise revenues by ending some tax credits and deductions that benefit the wealthy and big corporations.
"While on our side we're scouring the federal budget for responsible savings, we also have to scour the bloated tax code, and we need to close wasteful tax loopholes that only benefit the wealthiest Americans and biggest corporations," Murray said.
Meanwhile, Senator Barbara Mikulski, who heads her chamber's Appropriations Committee, also has called for a two-year budget deal to ease the path for normal spending bills.