WASHINGTON — An FBI lawyer’s cryptic reference to a “secret society” during a text exchange with a colleague has given Republicans fresh ammunition in their apparent effort to cast doubt on special counsel Robert Mueller’s investigation into possible Trump campaign ties to Russia. It comes as congressional investigators increasingly turn their focus to the conduct of top federal law enforcement officials.

GOP lawmakers and conservative media have seized on the notion of a coordinated anti-Trump effort within the FBI since candid conversations emerged last month between the FBI lawyer, Lisa Page, and Peter Strzok, the FBI agent with whom she was romantically linked.

Wisconsin Republican Sen. Ron Johnson, head of the Homeland Security and Government Affairs Committee, focused relentlessly on a reference to the “secret society” he said came from Page, and linked it with what he said was a whistleblower’s account that there had been “secret off-site meetings” involving “a number of high level, FBI officials,” though he declined to identify the officials involved or whether the potential whistleblower worked at the FBI.

Johnson released two pages out of the 384 pages of texts between Strzok and Page that the Justice Department turned over to Congress last Friday, though none included the “secret society” reference he has mentioned publicly.

But a separate congressional Republican source provided the exchange in question to NBC News. Sent within hours after Mr. Trump won the presidency, it shows Strozk and Page expressing concern in texts about what lay ahead.

“Seems kind of depressing,” Page wrote Strzok. “Maybe it should just be the first meeting of the secret society.”

The source told NBC News that it was the only explicit reference to a “secret society” in the most recent Justice Department disclosure.

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Johnson backtracked somewhat on Wednesday, saying he had merely “heard” about the existence of a secret society and did not have direct evidence of such a rump organization within the FBI.

“All I said is when I read those in those texts, that’s Strozk and Page’s term,” Johnson said when pressed by reporters on Capitol Hill on whether he believed such a group existed. “I have heard there was a group of managers in the FBI that were holding meetings offsite. That’s all I know.”

The communications, requested by House Intelligence Committee Chairman Devin Nunes, covered periods when both Strzok and Page worked on the investigation into Hillary Clinton’s use of a personal email account as secretary of state and when Strzok was assigned to Mueller’s team. Strozk was removed from the Mueller probe soon after the former FBI director assumed control of it, after discovering disparaging comments Strzok had made about Mr. Trump.

Republicans say the messages that Johnson disclosed publicly and others shared with NBC News present a troubling pattern of bias.

In another exchange last May, just after Mueller was named to oversee the Russia probe following Mr. Trump’s termination of former FBI Director James Comey, the two speculate about whether or not to participate.

“For me, and this case, I personally have a sense of unfinished business,” Strzok writes. After referring to the bureau’s code for the Clinton email probe, he adds: “Now I need to fix it and finish it.”

Other messages, including a smaller batch released in December, showed the two to be harshly critical of Mr. Trump during his campaign, while newer messages appear to show the pair’s personal political views affecting their investigatory work.

The Justice Department declined to comment on the contents of the new texts. Strzok and Page have also had no comment about the texts.

Democratic lawmakers are largely not addressing the specific exchanges in the new text messages, but instead warning of the lasting damage Republicans are doing to U.S. law enforcement institutions with what they view as a political campaign to undermine the Mueller probe.

“These Republican attacks show their desperation at the fact that Mueller already has obtained two guilty pleas, two indictments, and at least two cooperating witnesses,” Reps. Jerrold Nadler, Elijah Cummings and Adam Schiff, the top Democrats on the House Judiciary, Oversight and Intelligence Committees, said in a joint statement.

In a letter to the new FBI Director, Christopher Wray, Johnson highlighted three other exchanges he suggested raised new questions the impartiality of the Clinton investigation. In one from May 4, 2016, just after Mr. Trump all but locked up the Republican nomination, Strzok writes: “Now the pressure really starts to finish MYE” — bureau shorthand for the Clinton probe.

On June 30 of that year, Strzok shares with Page that a draft of Comey’s public statement announcing the bureau would not criminally charge Clinton would remove a specific reference to Clinton emailing President Obama from her personal email account, changing it only to say she emailed “another senior government official.”

Nadler and Schiff say any suggestion that FBI officials tilted their investigation to favor Clinton overlooks the fact that the same officials — including Comey — had directly notified lawmakers within weeks of the election that they were reopening the probe, “an unprecedented action that severely damaged Clinton’s bid for President.”

The broader point, according House Oversight Committee Chairman Trey Gowdy, R-S.C., was that the overwhelming theme of exchanges between Strzok and Page was a clear anti-Trump bias which could weaken the bureau’s standing if and when any cases that stem from the probe go to trial.

“The process and the result have got to be above reproach,” Gowdy said. Bias issues are “always relevant and judges give you the widest latitude. It destroys a jury’s ability to have confidence if they think you’re biased. It just does.”

Republican lawmakers, and now Mr. Trump himself, are pressing the Justice Department to explain why there was a five month gap in the text messages provided to Congress. Wray said in a memo delivering the messages to lawmakers that texts between Dec. 14, 2016 and May 17, 2017 were not preserved “due to misconfiguration issues related to rollouts, provisioning, and software upgrades.”

Johnson pressed Wray in his letter to detail the scale and scope of records lost and whether there are other records of communications. GOP Sen. Richard Burr, chairman of the Senate Intelligence Committee, said though he was “only concerned about the things that may affect our investigation, and we're satisfied that we have what we need.”