House and Senate negotiators have unveiled their budget blueprint to partially undo the sequester, but ardent defenders of 2011’s automatic spending cuts sought to undercut the agreement even before it was announced.

Rep. Paul D. Ryan, R-Wis., and Sen. Patty Murray, D-Wash., heralded an agreement on Tuesday evening to increase the discretionary spending cap for fiscal 2014 to the neighborhood of $1 trillion, an increase from the $967 billion under the required sequester created by the 2011 debt limit deal. The new deal includes increases in spending allocations on both the security and non-security sides of the ledger, and the agreement would last for two years.

Ryan said the measure “reduces the deficit by $23 billion and it does not raise taxes and it cuts spending in a smarter way.” He added, “This agreement makes sure that we don’t have a government shutdown scenario in January.”

Murray lauded the agreement as a break in the gridlock that has gripped the legislative branch in recent years, calling it an “important step in helping to heal some of the wounds here in Congress.”

But Senate Minority Leader Mitch McConnell, a key architect of the 2011 budget framework, continued to tout the 2-year-old Budget Control Act as a success, underscoring the delicacy of any new deal to put in place a bicameral agreement on spending.

“I have said in the past I remain convinced that the Budget Control Act has done what it was supposed to do. We’ve reduced government spending for two years in a row, for the first time since right after the Korean War,” the Kentucky Republican said. “Many of us came to Congress to do just that.”

McConnell added: “I think it has been a success, and I hope we … don’t revisit it.”

To be sure, any agreement would likely have undone the sequester cuts in some way. The original Budget Control Act never intended for sequestration to take effect; the 2011 supercommittee was tasked with finding more targeted cuts and deficit reduction but failed to do so — which caused the automatic spending cuts to kick in.

The middle ground for this year’s deal-makers includes fee increases and other sources of revenue that don’t qualify as tax increases (such as higher contributions by government employees to their pensions). That’s far short of the broad tax increases Democrats sought and the major entitlement cuts Republicans wanted.

Some outside conservative groups were criticizing the framework even before its release for allowing any spending increases, even with the offsetting reductions and receipts.

“Republicans should once again stand firm in upholding the modest sequestration spending cuts that both parties agreed to for the current fiscal year,” Americans for Prosperity President Tim Phillips said in a statement. “Otherwise, Congressional Republicans are joining liberal Democrats in breaking their word to the American people to finally begin reining in government over-spending that has left us over $17 trillion in debt.”

The influential Koch brothers, who have funded super PACs for Republicans and against Democrats, also issued an open letter to Congress urging lawmakers to stick to the $967 billion level.

Those statements came as 33 House Republicans called for the House to move forward with a continuing resolution at the post-sequester level of $967 billion, while stressing it was a fallback position.

“The letter is not, ‘What are we going to vote for, what can we support?,’” lead signatory Mick Mulvaney, R-S.C., told CQ Roll Call in a phone interview. “All we’re saying is, ‘Look, if we don’t get anything we can support, we are not going to tolerate a government shutdown.’”

It is not yet clear whether the conservative backlash against the looming budget deal would be enough to scuttle it in the House or Senate.

Senate Majority Whip Richard J. Durbin said that among Senate Democrats,“I think there is a positive feeling” about the potential deal. But he also noted that some objections came up during a caucus lunch Tuesday.

While he declined to get into many details, the Illinois Democrat said Sen. Jack Reed, D-R.I., spoke up about trying to get an extension of expanded unemployment benefits. Earlier in the day, Durbin stressed that not getting an unemployment insurance extension in the package shouldn’t derail the entire deal.

“It’s important to extend unemployment, but I don’t think we want to shut down the government over this issue. We have to reach a point where we’re prepared to give and ask the same of Republicans,” Durbin said.

Rep. Steve Israel of New York, the chairman of the Democratic Congressional Campaign Committee, said that the House Democratic Steering and Policy Committee was reviewing what he called a menu of items in the most recent iteration of a Ryan-Murray deal, which House Budget ranking member Chris Van Hollen, D-Md., brought to the group. Israel reiterated the concerns of many Democrats about leaving unemployment insurance out of the deal and the possibility of increased federal pension contributions, but he wouldn’t draw lines in the sand.

Lawmakers on both sides of the Rotunda representing a significant number of federal workers have sounded particularly skittish about those provisions, including Van Hollen and Sen. Benjamin L. Cardin, D-Md., who said in a Monday letter that federal workers faced a “relentless attack.”

Emma Dumain and Meredith Shiner contributed to this report.