Republicans stepped up their calls Thursday for Attorney General Loretta E. Lynch to recuse herself and appoint a special prosecutor to investigate Hillary Clinton after Ms. Lynch admitted to meeting earlier this week with former President Clinton at the Phoenix airport.

Ms. Lynch has insisted the meeting was impromptu and touched on personal issues, not the FBI’s criminal investigation into Mrs. Clinton’s email practices at the State Department. And Democrats rushed to defend Ms. Lynch, saying that despite the unforced error, she is capable of overseeing the investigation.

But Sen. John Cornyn, Texas Republican, said Ms. Lynch needs to remove herself from oversight.

“This incident does nothing to instill confidence in the American people that her department can fully and fairly conduct this investigation, and that’s why a special counsel is needed now more than ever,” the Texas Republican said.

Presumptive GOP presidential nominee Donald Trump questioned the judgment of both Ms. Lynch and Mr. Clinton, who is the husband of Mr. Trump’s likely opponent in the November election.

“You know, I’ve been talking about the rigged system — how it’s rigged,” Mr. Trump said on “The Mike Gallagher Show.” “And this is terrible, and nobody can understand why nothing’s happened. And you see a thing like this, and even in terms of judgment, how bad a judgment is it for him or for her to do this?”

Mrs. Clinton has said she made a bad choice in using a secret email account tied to a server she kept at her home in New York, which effectively shielded her emails from public view for nearly six years. Hundreds of those messages have since been deemed classified, and authorities are probing the arrangement.

The FBI has said it won’t be rushed to judgment in the investigation, but President Obama has faced criticism for saying he didn’t think Mrs. Clinton jeopardized America’s national security. Mr. Obama has also endorsed Mrs. Clinton, raising questions about whether that would skew the investigation.

Ms. Lynch’s meeting with Mr. Clinton added fuel to complaints that the attorney general has at the very least created the appearance of a conflict of interest in the event that the FBI recommends a criminal prosecution for Mrs. Clinton but the Justice Department decides not to pursue the case.

Judicial Watch, a conservative legal group that has worked to secure the release of records tied to Mrs. Clinton’s email set-up, said on Thursday that the Justice Department’s inspector general should investigate the meeting.

“Attorney General Loretta Lynch’s meeting with Bill Clinton severely undermined the already low public confidence in her agency’s criminal investigation of Hillary Clinton,” said Judicial Watch President Tom Fitton.

The White House deflected questions on the propriety of the meeting but said Ms. Lynch understands the importance of an independent Justice Department.

“The president’s view is that this is an investigation that should be conducted free of any sort [of] political interference,” said White House press secretary Josh Earnest. “And the attorney general has indicated that that’s exactly her expectation as well.”

Ms. Lynch said the conversation with Mr. Clinton was primarily social, and that the former president came over and talked about his grandchildren and his travel.

“So that was the extent of that, and no discussions were held [on] any cases or anything of that [nature]. And he didn’t raise anything about that either,” Ms. Lynch told reporters.

Senate Minority Leader Harry Reid, Nevada Democrat, said on Thursday that he can’t control who meets with whom but that Ms. Lynch’s ethics are “above reproach.”

Sen. Christopher A. Coons, Delaware Democrat, likewise defended Ms. Lynch, but said in an appearance on CNN that her meeting with Mr. Clinton sends the wrong message.

“I think she should have steered clear, even of a brief, casual social meeting with the former president,” Mr. Coons said.

The meeting also took place the same week that a House committee released its long-awaited report on the run-up to and aftermath of the Sept. 11, 2012, terrorist attack on a U.S. compound in Benghazi, Libya.

The report did not reveal any conclusive damning information about Mrs. Clinton, who was secretary of state at the time, but it detailed evidence the panel collected about security preparations in the run-up to the attack and why resources might not have been deployed more quickly during the attack itself.

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