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“How do you boil down the 10,000 other cases to that one [guilty] guy?” he asked, before adding. “And what do you do about the false positives?” referring to innocent people wrongly suspected, arrested or interrogated.

Mr. Graham said Sunday and Monday that a foreign agency’s suspicion should be enough to mark an U.S. resident for federal investigation, but Mr. Mudd noted that Russian intelligence and security agencies have waged war for two decades with Islamic extremists in the North Caucasus and “have an interest in getting us to chase Chechens in this country.”

However, even Internet postings are not sufficient evidence to open an investigation, Mr. Mudd said.

“If it’s looking at websites, I can tell you that’s protected [free] speech,” he said. “There’s only so much you can do in an open society.”

But Mr. Graham also issued a stark warning, saying lawmakers must make sure they have the strongest laws on the books because there will be future attacks on American soil.

“You knew this day was coming and there are more days like this coming,” Mr. Graham told reporters on Capitol Hill. “I can tell you, I hate to say it, but there are more of these people coming after us, and the new way of hitting American is do to it inside, and there’s efforts afoot by international terrorists to recruit American citizens for their cause and we can’t be right every time. So, this is the future and we need to get ready for it.”

• Sean Lengell and Seth McLaughlin contributed to this report.