There are, believe it or not, grounds for hoping that the sequester, stupid as it is, might open the way to ending our nation’s budget stalemate.

Hope is in short supply right now, but the case for seeing a way out of the current mess rests on knowable facts and plausible assumptions.

It starts with the significant number of Republicans in the Senate -- possibly as many 20 -- who think what’s going on is foolish and counterproductive. The White House is betting that enough GOP senators are prepared to make a deal along lines that President Obama has already put forward.

Obama’s lieutenants argue that while Republicans are aware that the president is seeking new revenue through tax reform, many did not fully grasp the extent to which he has offered significant long-term spending cuts. These include reductions in Medicare and a willingness (to the consternation of many Democrats) to alter the index that determines Social Security increases. Obama has proposed $930 billion in cuts to get $580 billion in revenues.

Senior administration officials note that Obama cannot stray too far from his existing offer, which was already a compromise, without losing the Democratic votes a deal would need. But his framework, they believe, could create a basis for negotiation with Republican senators such as Sen. Lindsey Graham, who dislike the deep automatic cuts in defense spending, and others, such as Sens. Susan Collins and Bob Corker, who dislike government-by-showdown.

Graham was especially bullish, declaring that Obama’s outreach to Republicans — the president invited about a dozen GOP senators to dinner on Wednesday night — was “the most encouraging engagement on a big issue I’ve seen since the early years of his presidency.”

If the Senate actually passed a bipartisan solution, it would still have to clear the House, requiring Speaker John Boehner to allow yet another bill get through with a large number of Democratic votes. But the sequester almost certainly marked the high point of solidarity among House Republicans. Letting it take hold was an easy concession for Boehner to make to more militant conservatives, and kept them from pushing toward government shutdowns or a politically and economically dangerous confrontation over the debt ceiling.

Now comes the hard part for Boehner. Already, there is pushback from more moderate conservatives against the depth of the budget cuts that Rep. Paul Ryan will have to propose in in order to balance the budget in 10 years. At least some House Republicans may come to see a bipartisan Senate-passed deal as more attractive than the alternatives.

In the meantime, the House passed a continuing resolution on Wednesday to keep the government functioning until the fall. It provided the Defense Department with greater flexibility to handle the automatic cuts. The Senate and the House must agree on a bill by March 27, and Sen. Barbara Mikulski, a Democrat who chairs the Appropriations Committee, argues that flexibility on defense should be matched on “compelling human priorities” and in other domestic areas, including science and technology. Yet both sides seem committed to keeping the government open, which would create a kind of cold peace.

But this will not prevent the automatic cuts from becoming more noticeable, and they are likely to get more unpopular as time goes on. Their effects will be felt by, among others, air travelers, school districts, state governments, universities, and the employees of defense and other government contractors. House Republicans representing districts with large military bases are apt to be especially eager to reverse the cuts.

From Obama’s point of view, engaging with Senate Republicans now to reach a broad settlement makes both practical sense, because there is a plausible chance for a deal, and political sense, because he will demonstrate how far he has been willing to go in offering cuts that Republicans say they support. In the process, he would underscore that the current impasse has been caused primarily by the refusal of House Republicans to accept new revenues.

While it’s the GOP that has been using serial, self-created crises to gain political leverage, many in the party are no less worn out by them than the Democrats. “Even we are tired . . . of lurching from one cliff to another,” Sen. John McCain, R-Ariz., told The Wall Street Journal on Wednesday. “I think that’s lending some pressure towards trying to come up with some kind of a grand bargain.”

Thus does the strongest reason for hope arise from one of the most basic human responses: exhaustion.