But the witnesses sidestepped Mr. Grahams question, saying only that they were working on his request. That provoked an angry intervention from the committee chairman, Senator Charles E. Grassley, Republican of Iowa, who banged his gavel and told Mr. Graham, his voice rising, I want you to proceed until you get an answer.

Mr. Graham eventually ended his questioning without getting one. But later in the hearing, Senator Richard J. Durbin, Democrat of Illinois, suggested that the senators emotion at the thought that their government could invade their privacy and use the information against them was just part of the bigger picture.

What about the privacy of the Americans who are not in this room? he asked.

The warrantless surveillance program traces back to President George W. Bushs Stellarwind program, introduced after the Sept. 11, 2001, attacks. Stellarwind permitted the National Security Agency to wiretap Americans international phone calls without the court orders required by the Foreign Intelligence Surveillance Act, or FISA, of 1978.

After it came to light, Congress legalized a form of the program in 2008 with the FISA Amendments Act. It permits the government to collect, from American internet or phone providers and without warrants, the communications of foreigners abroad who have been targeted for any foreign intelligence purpose even when they are talking to Americans.

Privacy advocates want Congress, as part of any bill extending the law, to require warrants before officials may use Americans identifiers, like their email addresses, to search the repository of messages previously collected by the program. But Stuart J. Evans, a top intelligence official at the Justice Department, testified on Tuesday that imposing such a limit would grind the entire FISA process to a halt because investigators need to quickly search a large volume of such queries to process leads, and because such queries are typically undertaken at an early stage, when investigators have not yet found evidence to establish probable cause of wrongdoing.

Several lawmakers also pressed the officials about a decision by Dan Coats, the director of national intelligence, to shelve an N.S.A. effort to estimate how much incidental collection of Americans information the program sweeps up. Bradley Brooker, the acting general counsel to Mr. Coats, said that systematically determining who is using email accounts that are not of foreign intelligence interest would invade peoples privacy and divert resources.

To underscore their message that the program is too valuable to curtail, Mr. Brooker and other officials disclosed several additional examples where the program had been useful. They included detecting an unidentified country that was smuggling goods in violation of sanctions, and finding someone in Western Europe who was talking to a member of the Islamic State about purchasing material to build a suicide belt.

Mr. Ghattas said the government had used the program to investigate Shawn Parson, a Trinidadian social media propagandist for the Islamic State whose network distributed prolific amounts of English-language recruiting pitches and calls for attacks before he was killed in Syria in August 2015.

The F.B.I. had been investigating Mr. Parson since October 2013 based on his online postings, Mr. Ghattas said, and information it shared from that collection with unspecified allies had helped them identify other Islamic State supporters and had potentially prevented attacks in those countries.

Follow Charlie Savage on Twitter @charlie_savage.

A version of this article appears in print on June 28, 2017, on Page A14 of the New York edition with the headline: Up-and-Down Hearing On Surveillance Program.

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