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So far, none of them has called for Mrs. Sebelius to lose her job over the situation. Republicans have not been so shy.

Mr. Roberts — who knows Mrs. Sebelius personally from their shared home state of Kansas — and co-signers of his letter outlined a litany of problems with HealthCare.gov, a website that was supposed to connect residents of 36 states with health plans on the individual market but has frequently crashed since its Oct. 1 debut.

The online portal has struggled to transmit accurate reports to insurers, and doubts about the site’s technical safeguards have persisted despite the Obama administration’s pledge that the system is secure.

Mr. Obama has offered no indication he will oust Mrs. Sebelius. Instead, the administration says it is focused on making HealthCare.gov work for the “vast majority” of users by the end of the month.

Mrs. Sebelius, who said she is accountable for Obamacare’s rollout failures but refused to resign, will travel to Atlanta on Friday to promote Obamacare and meet with in-person assistants who are helping people enroll in health coverage. Her trip follows Mr. Obama’s jaunt to Texas on Wednesday to urge reluctant state officials to expand Medicaid enrollment to people making up to 138 percent of the federal poverty level.

Mrs. Sebelius is coming off a pair of bruising hearings on Capitol Hill, where she doubled down on the importance of the health care law for sick Americans who may gain insurance for the first time.

But Republican lawmakers say the reforms are fundamentally flawed. Millions of Americans are losing health care plans that do not meet minimum-coverage standards under Obamacare, and they cannot explore alternatives if the online portals are not working.

These customers might be able to find better coverage at a better price under Obamacare’s exchanges, which offer income-based subsidies to defray premium costs, but they would have to sign up by Dec. 15 to have coverage before next year.

“These people could be left without any insurance at all if the exchange systems are not working soon,” Mr. Roberts and fellow lawmakers wrote.

Julie Bataille, spokeswoman for the Centers for Medicare and Medicaid, said Thursday she thinks the website will run well enough by late this month to give consumers enough time to enroll. She also downplayed talk of extending Obamacare’s six-month enrollment period past March 31.

“That is certainly a long window of time for individuals to get information, get educated and make their plan selection by the end of March,” she told reporters on a conference call.

CMS has until the end of business Friday to respond to a subpoena from House Ways and Means Committee Republicans that demands figures on how many people have enrolled so far under Obamacare, but the administration has said repeatedly that it will not release the data until “mid-November.”

A committee spokeswoman said Chairman Dave Camp, Michigan Republican, had not received anything as of Thursday afternoon.

White House spokesman Jay Carney said the subpoena appeared to be a political ploy designed to twist the knife over dismal enrollment.

“If the purpose is to point out, which I’m sure it is, that enrollment numbers will be low for October — take it from me, they’ll be low in October,” he said.