By contrast, the split among Republicans appears to be widening.
On Tuesday, Meehan was one of several who publicly urged the party leadership to pass a simple, short-term spending bill — a continuing resolution, or CR, in legislative jargon — without controversial add-ons, as the Senate did.
"We fought the good fight. Time for a clean CR," said another member of the group, Rep. Scott Rigell of Virginia.
Many hard-core conservatives, who have out-fought the party's more centrist members for years, are skeptical. House Speaker John A. Boehner (R-Ohio) can't afford to capitulate now, they say.
"We have certain times in the calendar year when we have must-pass bills. This is one of them," said Rep. Marlin Stutzman (R-Ind.). "For John Boehner, he goes this far and doesn't walk out with something, it doesn't look good. He has to get something."
Since taking the speaker's job, Boehner has operated in an environment in which his most conservative members have been able to exercise what in effect has been a tea party filibuster in the House.
Republicans hold a 232-to-200 advantage in the chamber. That's the second-largest GOP majority in generations. But because Democrats vote in near unison on high-profile legislation, as few as 16 Republican defections can defeat bills offered by the majority or prevent them from reaching the floor.
Such dissent has most often come from the right. Pressure from conservatives in the House forced Boehner and his leadership colleagues to embrace the conservatives' anti-Obamacare approach to the spending battle.
As the likelihood of a shutdown increased over the weekend, however, the marginalized moderates began to stir.
In a closed-door meeting Saturday, Rep. Charlie Dent (R-Pa.) was the lone member to openly urge the leadership to consider other options. He gathered with a group of like-minded colleagues to discuss whether they could bring pressure to bear from the center.
He and Rep. Peter T. King (R-N.Y.) sought to round up 40 colleagues to insist upon a resolution, enough to provide a cushion in case some got cold feet. King thought they had as many as two dozen committed votes. But on a key procedural vote Monday, Dent and King were the only moderates to vote no.
King and others attributed their failure to members of their party being unwilling to vote publicly against a provision involving insurance premiums for White House and congressional staff members. The bill would have barred the government from paying part of the premiums. Even though most large employers pay part of the premiums for their workers, the "subsidy" for congressional staffers has become a hot-button issue among conservatives.
"If it was a secret ballot, you'd have three-fifths, two-thirds, voting to end it all," King said. "On this one, it was particularly hard because it looked like if you were voting no, you were voting to protect a privileged class. That's the way it would be played in a primary, which is what Ted Cruz people want to do — run primaries against Republicans."
As the standoff drags on, however, the Republican centrists say their ranks will grow and eventually prevail.
"The most important thing for the country is to get a conclusion," said Rep. Frank A. LoBiondo (R-N.J.), whose district posted the third-largest margin of victory for Obama in last year's presidential election of any Republican-held district. "The politics of it are tough, and I get singled out. But I want to do the right thing for the country."