Supporters of Mrs. Clinton have argued that the committee’s mission has crept far beyond its original scope: to investigate the Benghazi attacks, which killed four Americans, including Ambassador J. Christopher Stevens. Republican committee members have said that they are within their right to look into her email use because the resolution that created the panel directed them to examine how the administration complied with previous inquiries into the attacks. Mrs. Clinton’s emails relating to the attacks were not handed over to any of the panels conducting those inquiries.

Other panels in Congress may consider investigating the matter. Mr. Graham, who oversees a Senate subcommittee with sway over the State Department’s budget, said that the department “seems to have a system that is not working very well” in regards to its production of documents to Congress.

“I’m going to ask them whether they think Mrs. Clinton has handed over everything she should and what they are going to do about it,” he said. “And if they give me runaround responses, we’ll drag them up on Capitol Hill and make them answer these questions in public.”

While the State Department acknowledged that it did not have several of Mrs. Clinton’s emails, it also told the Benghazi committee that it had not turned over other messages of hers. The department said that it had not done so because the contents of those messages fell outside the requests made by the committee.

“The State Department is working diligently to review and publish the 55,000 pages of emails we received from former Secretary Clinton,” it said in a statement.

That statement is unlikely to satisfy the committee, which believes it has been clear in its requests. Members of the panel have contended that the State Department has withheld documents to protect Mrs. Clinton and grind the investigation to a halt. State Department officials have said that one of the reasons it has taken so long to produce documents is that the department’s record-keeping system is cumbersome. They have also said that the committee has not been specific enough in its requests.