Two top Senate investigators floated the idea of immunity Tuesday to Bryan Pagliano, the staffer who set up the email server in former Secretary of State Hillary Rodham Clinton’s home, in exchange for testimony about her activities.

Mr. Pagliano last week said he would assert his Fifth Amendment right against self-incrimination in refusing to answer questions about the matter, but Sens. Chuck Grassley and Ron Johnson, the chairmen of two key committees investigating the situation, said they have authority to extend immunity if Mr. Pagliano is willing to talk.

They proposed a proffer session, where Mr. Pagliano would let them know what he knows in an off-the-record setting, and all sides could then decide what steps to take.

“The Committees have the authority to obtain an immunity order, to acquire the information they need, while also protecting your right against self-incrimination,” the two senators wrote in a letter to Mr. Pagliano’s lawyer.

The senators would need to coordinate with the Justice Department in order to secure an immunity deal.

Their offer raises the stakes for Mr. Pagliano and could create more peril for Mrs. Clinton, who has insisted she complied with all laws when operating the server.

But after days of refusing to apologize for setting up her own server, Mrs. Clinton reversed course Tuesday and told ABC News that she was “sorry.”

“In retrospect, as I look back at it now, even though it was allowed, I should have used two accounts — one for personal, one for work-related emails. That was a mistake. I’m sorry about that. I take responsibility,” Mrs. Clinton said in the TV interview.

Pressed to clarify whether she made a mistake in setting up a private email account and private server to conduct official business, Mrs. Clinton responded: “I did. I did.”

“As I said, it was allowed, and there was no hiding it. It was totally above board. Everybody in the government I communicated [with] — and that was a lot of people — knew I was using a personal email,” she said. “But I’m sorry that it has, you know, raised all of these questions. I do take responsibility for having made what is clearly not the best decision.”

Mrs. Clinton repeated much of what she told ABC News — sometimes using the same words — in a Facebook entry posted Tuesday evening, telling users of the world’s biggest social-media outlet at the start that “I wanted you to hear this directly from me.”

Mrs. Clinton’s acts of contrition comes as her campaign acknowledged that she must rework her image and try to regain her footing in the race. She’s lost ground in the polls, and a majority of voters say they don’t trust her, mostly due to the email controversy dogging her campaign.

While offering an apology, Mrs. Clinton continued to insist that the email arrangement and the way she used it was “allowed” by the State Department, as she did in interviews Friday with NBC News and Monday with The Associated Press.

Security experts have disputed Mrs. Clinton’s claim that she followed the law, saying that she handled hundreds of classified messages on the server.

Mrs. Clinton has maintained that none of the messages were secret at the time she sent or received them, and were only upgraded now that they are being released to the public. The New York Times reported Tuesday, however, that a new review by the inspector general for the intelligence community has concluded two messages did contain information that was top secret at the time.

The State Department contested that review, saying it was “premature.”

John Kirby, a department spokesman, said they are doing their own analysis and “it’s not uncommon or atypical for there to be this kind of give-and-take between agencies on something like this.”

The department, however, is reeling from the mess Mrs. Clinton left. It faces more than 30 open records lawsuits demanding the release of emails Mrs. Clinton and her top aides kept secret for years after they left government service.

Last week the Obama administration sought to halt most of those lawsuits, asking the federal court in Washington to appoint a single judge to decide what to release and how quickly to do it.

Judicial Watch, which alone accounts for 16 of the open records lawsuits, filed papers Tuesday opposing that move, saying the government hasn’t even said how many emails top Clinton aides Huma Abedin and Cheryl Mills turned over last month.

Facing embarrassing questions about its record-keeping, the State Department announced a new “transparency coordinator,” Ambassador Janice L. Jacobs, to try to get a handle on things. The RNC, however, said Ms. Jacobs has already contributed the maximum amount to Mrs. Clinton’s 2016 presidential campaign primary account, calling her independence into question.

Mr. Kirby said she won’t be involved in the classification decisions concerning Mrs. Clinton’s emails.

Mr. Pagliano was paid by Mrs. Clinton’s political operation through early 2009, then shifted over to the State Department, where he was an information technology specialist. At the same time, however, Mrs. Clinton’s campaign has said she paid him to maintain the server she kept at her New York home.

Mr. Kirby, the State Department spokesman, refused to say whether Mr. Pagliano was required by department rules to report outside income paid to him by Mrs. Clinton.

Senate Majority Whip John Cornyn, Texas Republican, said he was familiar with discussions about offering immunity to Mr. Pagliano to testify and backed the proposal.

“I think it sounds like a good idea,” said Mr. Cornyn, who also serves on the Senate Committee on the Judiciary.

“I think it is important to find out what the facts are, and I think that we don’t know what the facts are,” he said. “The FBI apparently is looking into it, but Congress [also] has … oversight responsibilities as well.”

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