Will the sequester get as much attention at town hall meetings in the coming week as immigration, job creation or the coming of health insurance exchanges? Will it get even half as much time at editorial boards and coffee klatches as the farm bill, the IRS affair or the fading debate over gun control?

More and more lawmakers in both parties are worried the answer may be a somewhat surprising “yes.” And they have nothing close to an easy answer for whether any more flexibility might be created inside Washington’s self-imposed spending straitjacket — aside from the possibility of reaching a bigger bargain on taxes and entitlements that would include repealing the sequester altogether.

The political pressure to ease the across-the-board nature of the situation, especially from middle-income independents, will only grow as spring gives way to summer.

National Park Service officials have ordered the blame-the-sequester signs taken off its visitor center doorways, but the cutbacks remain in effect for the surge of vacationers to experience — no more so than during peak tourist time in Washington. Hurricane season is starting before tornado season ends. But the deadly Oklahoma twister aside, the National Weather Service says there’s no way to shrink its 10 percent vacancy rate.

More than 115,000 workers at an alphabet-soup of agencies — including the IRS, EPA, HUD and OMB — were ordered to make Friday the unpaid start to their Memorial Day weekend. And the place with the most significant sequester challenge of all, the Defense Department, is on course to send each of its 650,000 civilian employees home for 11 days during the final three months of this budget year, starting the week after July Fourth.

Since March, when the sequester took effect, both parties have claimed their side is succeeding at making something of a political silk purse out of a budgetary sow’s ear. Republicans have sounded convinced that the indiscriminate nature of the cutbacks is proving their point that almost every office in government could do with a decent haircut. Democrats believe they are using the situation to persuade their constituents of the opposite — that taking 8 percent out of every domestic account and 5 percent off the top of every military program only highlights the need for the president’s “balanced approach.”

Both sides had been of the view that easing the sequester strictures would take away whatever pressure remains for reaching another deficit deal before the midterm elections.

But along the way, both sides agreed to make a couple of high-profile exceptions to quell protests from both influential industries and the millions of voters who rely on them. And the workarounds that have kept all the meat inspectors and air traffic controllers on the job have only intensified the interest of lawmakers in giving each agency similar permission to move money around while living under the same overall cutback.

Republicans are making the most noise about this at the moment. The roster is dominated by appropriators, including Susan Collins of Maine and Roy Blunt of Missouri in the Senate and Frank R. Wolf of Virginia in the House, but also includes such prominent deficit hawks as Sen. Patrick J. Toomey of Pennsylvania. They all argue the agencies should be forced to do what families do in setting priorities that fulfill the must-do list and ignore the nice-to-haves.

What’s been notable is the growing number of Democrats who say they agree, ranging from Montana’s Jon Tester and Colorado’s Mark Udall in the center of the Senate to D.C. Del. Eleanor Holmes Norton on the left margins in the House. Each has favorite programs or services they’d like to see relieved from the cuts, even at the expense of others, and all are confident like-minded political appointees in the administration would do their bidding if they only had the discretion.

If the bipartisan clamor for widespread flexibility grows after lawmakers return in a week, the groundswell might be rebuffed only with an unlikely agreement between GOP conservatives and the president. The Republicans already think President Barack Obama has too much power, and some will be loath to give him and his team any more. And the president has made clear he doesn’t want it.

One reason should be clear: Picking winners and losers is a double-edged sword, especially in a zero-sum game. Rewarding cancer research at the expense of a potential Crohn’s disease breakthrough, boosting Head Start over grants to high school STEM teachers or speeding development of a next-generation ship ahead of the newest night-vision goggles — such moves would make the president at least as many enemies as friends and would identify him as the “owner” of a process he’d still be arguing is fundamentally flawed.

It’s probably too late for Congress to come to a flexibility agreement that affects the current fiscal year. But the pressure will only grow — every month threatens the release of fresh indicators of the sequester’s drag on economic growth. And the problem does not go away Oct. 1. Given that this is another year where the annual appropriations process is at a dead end from the start, it’s safe to assume a continuing resolution bill will be keeping the government running for months into fiscal 2014.

Without a deal, agencies will be stuck where they are even though government inflation will have gone up another 2 percent.