(CNN) - Sen. Bob Corker on Sunday joined a few other Senate Republicans who say they're open to raising revenues as part of a deal on deficit reduction, adding there is a chance such an agreement could work.
House Speaker John Boehner, meanwhile, continued to maintain that increased tax revenue would not be part of the deficit reduction picture and sounded less hopeful on the idea of a "grand bargain."
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Corker said he could only see Republicans compromising on tax revenue if Democrats agree to big changes in entitlement programs, like Social Security and Medicare.
"I think Republicans, if they saw true entitlement reform, would be glad to look at tax reform that generates additional revenue. And that doesn't mean increasing rates. That means closing loopholes," the Tennessee Republican said on "Fox News Sunday."
As Washington grappled with how to lessen the impact of the forced spending cuts that went into place March 1, Republicans staunchly opposed the idea of bringing in more revenue, saying the real problem was on the spending side of the equation.
Some Republicans in the Senate, including Sen. Lindsey Graham, said they would be open to the idea of closing certain loopholes if it helped lessen the impact of the cuts on the military.
Both parties released their budgets last week. The House Republican budget reduces the deficit by $4.6 trillion over the next 10 years through spending reductions, while the Senate Democrats' budget proposes a $1.8 trillion deficit-reduction package, which includes a mix of spending cuts and tax increases.
Corker said Republicans want to see tax reform and a "75-year solution to entitlements." But, he argued, it all depends on how much Democrats are willing to shake up entitlement spending.
The senator is one of several Republicans who have been in touch with President Barack Obama during his recent outreach to Republicans. Obama visited Republicans on Capitol Hill last week and dined with 12 Republican senators, including Corker, the week before in order to discuss deficit reduction.
"I think, there, by the way, is a chance on a deal," Corker said. "I know the president is saying the right things, and we have an opportunity over the next four to five months."
Obama has said he's willing to propose changes on entitlements. But getting Democrats in Congress to agree with him may be an uphill battle.
Democratic Sen. Dick Durbin, however, said on Fox he believes Corker stated "a basic set of principles that both parties can rally around" and acknowledged that entitlement reform should be on the table.
"But there are ways to approach it - to reduce the cost of medical care, and still keep our promise to seniors across America," he said.
While Corker appeared open to tax increases down the road, Boehner said the conversation about tax revenue is finished. He said he was unsure whether a grand bargain could ever be reached.
"I don't know whether we can come to a big agreement. If we do, it'll be between the two parties on Capitol Hill," he said on ABC's "This Week." "The president got his tax hikes on January 1. The talk about raising revenue is over. It's time to deal with the spending problem."
He was referring to the tax rate increase for top earners that was included in the deal to avoid the fiscal cliff earlier this year.
Boehner has appeared somewhat skeptical of the president's recent "charm offensive" to Republican lawmakers in Congress and hasn't been shy in acknowledging the stark differences between the two parties. He said Sunday that while "it's always a good thing to engage in more conversation," the bottom line is the president favors "more taxes from the American people."
"We're not going to get very far," he said.
Asked if he trusts the president, Boehner said, "Absolutely."
"Oh, the president and I, as I have made very clear, have a very good relationship. We're open with each other, we're honest with each other, but we're trying to bridge some big differences," he said.