Budget brinkmanship is on tap again this fall, if this week’s renewed finger-pointing over a potential government shutdown is any guide.

Though Congress and the White House have just more than two months to strike a deal keeping the government running — and a little longer before the debt ceiling hits — they appear set to engage in a full-fledged messaging war over the August break before returning to the negotiating table in September.

Brendan Buck, a spokesman for Speaker John A. Boehner, R-Ohio, challenged the White House to take a shutdown off the table by dropping a veto threat on spending bills at the House budget’s austere level. That $967 billion level happens to be the one prescribed in the 2011 Budget Control Act that President Barack Obama signed into law, and it includes the full-year effect of the sequester that both sides have called bad policy.

It’s a $21 billion cut from this year’s spending level of $988 billion.

Senate Majority Leader Harry Reid, D-Nev., said it would be unacceptable to let the sequester continue beyond the Sept. 30 end of the fiscal year — and he vowed to oppose a stopgap spending bill that would do so.

“I think it would be a disaster for this country, and I would do everything within my ability to oppose that,” Reid said when asked about a continuing resolution at the $967 billion level.

White House Press Secretary Jay Carney largely dodged two questions about the threat of a shutdown. Instead, he talked about the need for a balanced approach to the budget and said the president does not support cutting education, innovation and infrastructure to “protect special interests.”

But he avoided getting pinned down on the central question: Will the president veto spending bills and risk a government shutdown to get his way on spending and taxes?

Buck nailed Carney for refusing to take a shutdown off the table.

“This scenario is a lose-lose for the American economy: either a tumultuous government shutdown or further job losses as a result of tax hikes,” Buck said. “And it cuts away all credibility the White House has when it decries — as it regularly does — such political tactics.”

Republican House appropriators have been eyeing a possible compromise spending level of $988 billion, though Senate Democrats are working off a much higher budget number assuming that the sequester goes away.

But despite weeks of negotiations between the White House, Senate Democrats and a group of compromise-minded Senate Republicans aiming at reviving some version of a grand bargain on taxes and entitlements, there has yet to be a breakthrough.

For their part, Senate Republicans continued to draw a hard line on the sequester — demanding that a stopgap bill stick to the House’s $967 billion level barring a larger budget deal.

“That’s the law. That’s what we have to stick to,” said Sen. Lamar Alexander, R-Tenn.

Senate Minority Leader Mitch McConnell, R-Ky., didn’t sound any more inclined to budge.

“My goal is to not walk away from the commitment we made on a bipartisan basis to the American people just two years ago to reduce spending, largely on the discretionary side, by $2.1 trillion over the next 10 years,” McConnell said.

While McConnell noted that he expected the White House to push for sequester relief, he rejected the idea of using new revenue to offset sequester cuts.

“I have, speaking for myself, no interest in reopening the subject of additional taxes,” McConnell said.

Some defense hawks, however, have seemed open to working with Democrats to come up with a way to undo the sequester cuts to national security. Sen. Lindsey Graham, R-S.C., said that he had discussed the subject with the other party as recently as Monday.

“Chit chat [about changing sequester cuts] becomes real when the threat gets better defined,” Graham said.

“The best thing that can happen for a guy like me is to make clear the consequence of sequestration and have that play out in terms of the world in which we live. It makes it harder to do Syria. It makes it harder to do Iran.”

Sen. Johnny Isakson, R-Ga., also said he is interested in a deal: “I would gladly swap sequester if we get entitlement reform. That’s the way you change sequester.”

But that’s exactly the kind of deal that has eluded Congress for more than two years.

Other Republicans, however, seem to be itching for a shutdown fight, implying it would be worth it if the end result was defunding Obamacare. For weeks, Sens. Mike Lee of Utah and Marco Rubio of Florida and others have been threatening to vote against any spending bills that fund the implementation of the 2010 health care law.

Reid so far is holding firm. A senior Senate Democratic aide suggested that if Republicans insist on keeping the sequester and refuse to compromise on revenue issues, they would own an ensuing shutdown.

“It would be a shame if Republicans shut down the government to keep a bad policy Ben Bernanke says is a key drag on economic growth right now,” a senior Senate Democratic aide said.

There remain doubts on Capitol Hill, however, on just how much resolve Obama will show on the issue.

The president agreed to sign a catchall spending bill in March even though it failed to lift the sequester — saying the country didn’t need another crisis. But that decision — along with the fiscal-cliff deal that has looked worse for Democrats with each passing day — effectively robbed the president and Democrats of leverage to force Republicans to the negotiating table.

Indeed, Republicans led by Boehner are back to demanding even deeper cuts in return for another increase in the debt ceiling, ignoring the president’s vow not to negotiate on that issue.

Carney repeated that vow Monday and Tuesday.

“We will not negotiate over Congress’ responsibility to pay the bills that Congress racked up. It is highly irresponsible to even flirt with that prospect,” Carney said.

He did say, however, that the president and the White House did want to “engage” separately with Republicans on a larger budget package.

Niels Lesniewski and Kerry Young contributed to this report.