Attorney General Loretta Lynch is facing a storm of criticism for her meeting--however impromptu--with former President Bill Clinton, and it remains to be seen whether the onslaught will have any impact on her role as the final arbiter in the FBI's investigation into Hillary Clinton, and whether she or her staff mishandled classified information through the use of a private email server while she was Secretary of State.

The watchdog Judicial Watch has called on the Justice Department inspector general to investigate the meeting between Lynch and Clinton because it "creates the appearance of a violation of law, ethical standards and good judgment," according to a release.

Judicial Watch also charged that the meeting "creates the broad public impression that "'the fix is in.'"

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The watchdog has also sued the State Department for the release of records on Clinton top aide Huma Abedin's employment at State, the Clinton Foundation and a consultancy. It has been deposing some of Clinton's top staffers from her tenure as secretary of state, including Abedin this week.

GOP presumptive nominee Donald Trump called the meeting between Lynch and Bill Clinton "one of the big stories of this week, of this month, of this year." He said on the "Mike Gallagher Show" that it was "terrible and nobody can understand why nothing's happened. And you see a thing like this and even in terms of judgment, how bad of judgment is it for him or for her to do this?" Later, in an interview with Sean Hannity that will air on Fox News Channel Thursday, Trump said he was "flabbergasted" by the meeting.

Sen. John Cornyn, R-Texas, said in a statement that the incident "does nothing to instill confidence in the American people that her department can fully and fairly conduct this investigation," and he called on her to appoint a special counsel "in light of the apparent conflicts of interest."

And Democrats, too, are concerned about the way the meeting appears. Former top Obama adviser David Axelrod tweeted that it was "foolish to create such optics."

I take @LorettaLynch & @billclinton at their word that their convo in Phoenix didn't touch on probe. But foolish to create such optics. — David Axelrod (@davidaxelrod) June 30, 2016

Sen. Chris Coons, D-Delaware, said in an interview on CNN that she should have told the former president, "'Look, I recognize you have a long record of leadership on fighting crime but this is not the time for us to have that conversation. After the election is over, I'd welcome your advice.'"

Lynch, for her part, has downplayed the meeting with Clinton.

The meeting occurred around 10:30 p.m. PT. The Attorney General had been traveling all day -- to Baltimore for an address about youth violence and gangs, and then to Phoenix to meet with police and top officials to promote community policing.

Bill Clinton was in Phoenix to do some fundraising, an aide told CBS News. Clinton also attended a meeting with roughly ten elected officials and community leaders. According to John Gomez, a vice president of Cartwright School District who attended the meeting, the group met at the office of a local law firm for about 45 minutes and discussed Arizona's role in the general election.

Hillary Clinton's campaign did not respond to a request for more information about the meeting.

The conversation between the former president and attorney general was, according to Lynch, "primarily social."

"Actually, while I was landing at the airport, I did see President Clinton at the Phoenix airport as I was leaving, and he spoke to myself and my husband on the plane," Lynch told reporters Wednesday. "Our conversation was a great deal about his grandchildren. It was primarily social and about our travels. He mentioned the golf he played in Phoenix, and he mentioned travels he'd had in West Virginia."

An aide to Bill Clinton says that he spotted her on the tarmac, but CBS News has been told that she was in an unmarked plane. Lynch did, however, have an advance team on the ground that Clinton may have seen or spoken to.

CBS News' Hannah Fraser-Chanpong and Alexander Romano contributed to this report.