WASHINGTON — Under growing pressure, the administration refused repeatedly to state a position Tuesday on legislation formalizing President Obama’s oft-stated promise that everyone who likes 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 administration’s health care law.

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 agency 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.

Said Tavenner, head of the Centers for Medicare and Medicaid Services: “We are now able to process nearly 17,000 registrants per hour, or 5 per second, with almost no errors.”

She encouraged consumers to log onto the site, and said the administration had estimated that enrollments will total 800,000 by late 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.

Tavenner also sought to reassure lawmakers who expressed concerns about cybersecurity 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.

“Has this happened before?” Scott asked. “Can you guarantee that Social Security numbers … are secure? Will you shut down the website, as my friends from the left have already suggested, until security issues are fixed?”

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 cancelled plans, which fall short of the coverage requirement under the health care law. One Democrat, Sen. Mary Landrieu of Louisiana, has proposed requiring insurers to do so.

But the Republican leader, Sen. Mitch McConnell of Kentucky, said Democrats had voted unanimously against similar proposals in the past and were having “foxhole conversions.”

“I think what will be really interesting to see in the Senate is the number of Democrats in very red states who are up in ‘14 and what they start demanding … in terms of adjustments to this law,” he said.

At the White House, press secretary Jay Carney refused repeatedly to state a position on the proposals, saying he hadn’t “reviewed or seen an examination internally” on any of them.

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.