WASHINGTON — With big, automatic budget cuts about to kick in, House Republicans are turning to mapping strategy for the next showdown just a month away, when a government shutdown instead of just a slowdown will be at stake.

Both topics are sure to come up at the White House meeting Friday between President Barack Obama and top congressional leaders, including Republican House Speaker John Boehner. A breakthrough on replacing or easing the imminent across-the-board spending cuts still seems unlikely at the first face-to-face discussion between Obama and Republican leaders this year.

To no one’s surprise, even as a dysfunctional Washington appears incapable of averting a crisis over economy-rattling spending cuts, it may be lurching toward another over a possible shutdown.

Republicans are planning for a vote next week on a bill to fund the day-to-day operations of the government through the Sept. 30 end of the 2013 fiscal year — while keeping in place the new $85 billion in cuts of 5 percent to domestic agencies and 8 percent to the military. The need to keep the government’s doors open and lights on requires the GOP-dominated House and the Democratic-controlled Senate to agree. Right now they hardly see eye to eye.

The House GOP plan, unveiled to the rank and file on Wednesday, would award the Pentagon and the Veterans Administration with their line-by-line budgets, for a more-targeted rather than indiscriminate batch of military cuts, but would deny domestic agencies the same treatment. And that has whipped up opposition from veteran Democratic senators on the Appropriations Committee. Domestic agencies would see their budgets frozen almost exactly as they are, which would mean no money for new initiatives.

“We’re not going to do that,” said Sen. Tom Harkin, D-Iowa. “Of course not.”

Any agreement needs to pass through a gantlet of House tea party conservatives intent on preserving the across-the-board cuts and Senate Democrats pressing for action on domestic initiatives, even at the risk of creating a foot-tall catchall spending bill.

There’s also this: GOP leaders have calculated that the automatic cuts arriving on Friday need to be in place in order for them to be able to muster support from conservatives for the catchall spending bill to keep the government running. That’s because many staunch conservatives want to preserve the cuts even as defense hawks and others fret about the harm that might do to the military and the economy. If the automatic cuts are dealt with before the government-wide funding bill gets a vote, there could be a conservative revolt.

Little to no progress has been made so far between House and Senate leaders and the White House, and there’s no guarantee that this problem can be solved, even though the stakes — a shutdown of non-essential government programs after March 27 — carry more risk than the across-the-board cuts looming on Friday.