Under growing pressure Tuesday, the administration refused repeatedly to state a position on legislation formalizing President Barack Obama’s oft-stated promise that people who like their existing coverage should be allowed to keep it under the new health care law.

Senate Democrats spoke dismissively of the proposals, signaling they have no intention of permitting a vote on the issue that marks the latest challenge confronting supporters of the Affordable Care Act.

An earlier controversy appeared to be ebbing on a law that has generated more than its share of them. Even so, one strong supporter of the health care law, Sen. Sheldon Whitehouse, D-R. I., good-naturedly told an administration official, “Good luck getting through this mess.”

Whitehouse spoke to Marilyn Tavenner, the head of the Centers for Medicare and Medicaid Services, which is deeply involved in implementing the law. She had assured lawmakers that initial flaws with the government’s website were systematically yielding to around-the-clock repair effort.

She said the administration had estimated that enrollments will total 800,000 by the end of November.

At the same time, she repeatedly refused to tell inquiring Republicans how many enrollments have taken place to date, saying that information would be made available at mid-month.

Across the Capitol, that reluctance drew a subpoena from Rep. Dave Camp, the Michigan Republican who chairs the House Ways and Means Committee. He said the material was “critical government information” that the administration has refused to provide voluntarily, and demanded that it be turned over by Friday.

Tavenner also sought to reassure lawmakers who expressed concerns about security at www.healthcare.gov.

Sen. Tim Scott, R-S.C., cited the case of a Columbia, S.C., attorney who used the website to look for coverage, only to learn later that some of his personal information had been made available to a different browser, a man in North Carolina.

Tavenner offered reassurance and said officials from her agency were attempting to get in touch with the man whose information had been disclosed.

The controversy over the ability of consumers to keep their existing plans flared last week, when insurance companies mailed out millions of cancellation notices, often citing the new health care law as the reason.

House Republicans intend to vote as early as next week on legislation that permits insurers to reinstate the canceled plans, which fall short of the coverage requirement under the law. One Democrat, Sen. Mary Landrieu of Louisiana, has proposed requiring insurers to do so.

In words Republican critics cite frequently, Obama pledged in mid-2009: “If you like your doctor, you will be able to keep your doctor, period” and “If you like your health care plan, you’ll be able to keep your health care plan, period. No one will take it away, no matter what.”

In recent days, Obama and top aides have sought to amend or clarify the pledge, a tacit acknowledgement that it hasn’t been kept.

Tavenner sidestepped questions on the subject, telling Sen. Lamar Alexander, R-Tenn., and other committee members she hadn’t read the legislation in question.

Sen. Johnny Isakson, R-Ga., displayed a sign behind his committee seat that read, “Tip of the iceberg.” On the sign’s iceberg, above the waterline, was the label ‘website failures.’ Below were examples of reported problems, including canceled coverage, higher co-pay and deductibles, premium increases and fraud and identify theft.