Senator Lindsey Graham, R-SC, says the Senate's chance of passing a budget deal hinges on one thing: its success in passing immigration reform.
"If we fail on immigration, there's no chance of a big budget bill," he said on CNBC's "The Kudlow Report." "If we pass immigration reform, maybe Congress can do two things at once and go to the budget."
A member of the bipartisan "Gang of Eight," Graham said the immigration legislation provides a compromise between border security and a path to citizenship for the 11 million undocumented immigrants living in the United States.
"Once you get on a pathway to citizenship, you have to pass through English exams, pay a fine, pay taxes, and get in the back of the line of the country where you came from," he said. "So it's an accommodation of border security and a pathway to citizenship linked together."
(Read More: Paul Ryan Sees House Passing 'Path to Citizenship')
Graham said he supports stricter border control but raised concerns over the price tag of Sen. John Cornyn's, R-TX, border security amendment.
"He's trying to improve border security," he said. "But here's one thing I want out of this bill: I want be deficit neutral. I don't want any more government programs that we've borrowed money from our kids and grandkids to pay for it. So right now, I believe the bill is paid for. If you want to add more border security, fine. Let's find a way to pay for it."
In addition to less borrowing, entitlement reform is a crucial factor in reducing the federal deficit, Graham said.
"To raise the debt ceiling to borrow a bunch more money, we're already $17 trillion in debt and climbing," he said. "I want fix the reason we're so much in debt. That's entitlement reform. We're going to have to adjust the age of retirement for Medicare to harmonize it with Social Security and adjust inflation rates because we're overpaying."
Graham offered a solution for Republicans on the budget deficit.
"Flatten the tax code, take some money out of the tax code that goes to the pensions and deductions, pay down some debt, and lower rates," he said.