WASHINGTON — The hard-line stance of Republican House members on the government shutdown is generating increasing anger among senior Republican officials, who say the small bloc of conservatives is undermining the party and helping President Obama just as the American people appeared to be losing confidence in him.

From statehouses to Capitol Hill, frustration is building and spilling out during closed-door meetings as Republicans press leaders of the effort to block funding for the health care law to explain where their strategy is ultimately leading.

“Fighting with the president is one thing,” said Senator Roy Blunt, Republican of Missouri. “Fighting with the president and losing is another thing. When you’re in the minority you need to look really hard to find the fights you can win.”

The complaints come from fervent opponents of the president’s health care overhaul, who say that the shutdown is overshadowing discussion of the problems associated with the law and ruining any chance for revising it.