Senate Majority Leader Mitch McConnell (R-Ky.), accompanied by Sen. John Barrasso (R-Wyo.), left, and Senate Majority Whip John Cornyn (R-Tex.), meets with reporters Tuesday on Capitol Hill. (Brett Carlsen/AP)

Senate Republicans celebrated on Tuesday after passing the first joint budget resolution in more than five years, but that success is just the start of the much more difficult task of funding the government.

Republicans said the next step will be for the Appropriations Committee to start drafting spending bills that meet the newly approved framework, which means cutting $496 billion in non-defense spending over the next decade. The budget adheres to domestic spending caps included in the 2011 Budget Control Act, also known as the sequester, and uses nearly $40 billion in off-budget funds to boost defense spending to over $563 billion.

No Democrat voted for the budget resolution, and President Obama has vowed to veto any funding bills that fall within the framework. Even as the budget passed, most Republicans acknowledged that it will be impossible to meet their own standards, though the party’s leaders insisted that they would try anyway.

“Our only tool is the funding process, and we’re going to try to do that,” Senate Majority Leader Mitch McConnell (R-Ky.) said in an interview. “It will be challenging, but we’re going to try to delay, rein in, restrict various bureaucratic overreach through the appropriation bills that spend the money.”

McConnell acknowledged that the process would start fights and would probably be difficult to pass, but his plan is to proceed with trying to advance deep-cutting spending bills. A spokesman for Senate Appropriations Committee Chairman Thad Cochran (R-Miss.) said the senator plans to start considering a dozen funding bills within a matter of weeks.

Republicans may be able to get through the committee process, but the chance of advancing legislation that closely adheres to their budget blueprint is nearly zero. Unlike in the House, where Republicans have enough votes to easily advance spending bills without the help of Democrats, the Senate will need the backing of at least six senators to bring their bills up for a vote.

Democrats vowed to block deep funding cuts for medical research, food stamps, housing programs for low-income workers and the federal Pell Grant program for students. Instead, they are calling on Republicans to help craft a plan to lift the spending caps for both domestic and military spending.

“We’re not going to sign on to a bill that goes to the sequester levels,” said Sen. Richard J. Durbin (D-Ill.), the minority whip. “There is no reason for us to support these funding levels on the domestic side.”

Republicans said Tuesday they will not waiver.

“We’re not going to wave the white flag on the day that we passed a budget agreement,” said Sen. Roger Wicker (R-Miss.). “I think a more positive can-do attitude is a good place to start.”

Despite Wicker’s sunny view, Sen. Charles E. Schumer (D-N.Y.) said he doesn’t expect the deepest-cutting funding bills to even make it out of committee. Schumer said he expects Republicans to refuse to vote in favor of cuts to programs that benefit their constituents.

“It wouldn’t get to the floor,” he said. “[Republicans] wouldn’t vote for it and Democrats wouldn’t vote for it.”

Democrats in Congress and the White House have said they want to see a dollar-for-dollar match in increased funding for domestic programs and the military before they’ll agree to any funding bills. House Speaker John A. Boehner (R-Ohio) said last month that he would be open to a conversation about lifting the spending caps in a deal similar to an agreement reached by former House and Senate Budget Committee chairs Sen. Patty Murray (D-Wash.) and Rep. Paul Ryan (R-Wis.) in 2011.

To date neither side has stepped forward to open negotiations. The White House has said it is waiting for Republicans to take the first steps.

“We welcome Speaker Boehner’s comments in support of a budget deal and look forward to seeing some more concrete steps from the Speaker and his Republicans colleagues on reversing sequestration for both defense and non-defense programs,” White House budget spokeswoman Melanie Roussell said in a statement last month.

The concern for some Democrats is that if Republicans go forward with their plan to advance spending bills ahead of a broader negotiation process, this could make the funding process even more complex.

“It complicates this immensely,” said Sen. Jack Reed (D-R.I.), a member of the Appropriations Committee. “In the first few bills you can move things around but there are vital functions that they would want to see funded and the arithmetic doesn’t work.”

Paul Kane contributed to this report.