A plan of action for how a second special counsel can investigate the FBI and Justice Department is spelled out in a point-by-point letter submitted by two Republican senators.

They specifically want to know whether former FBI Director James B. Comey lied to them and had a plan to double-cross President Trump.

Senate Judiciary Committee Chairman Charles E. Grassley of Iowa and Sen. Lindsey Graham of South Carolina sent a letter last week to Attorney General Jeff Sessions. They and two other Republicans want a special counsel with the powers of the Trump-Russia probe’s Robert Mueller to investigate suspected Justice Department corruption.

They attached a second letter to Michael E. Horowitz, the Justice Department’s inspector general, outlining all the avenues that need to be navigated — complete with new and expanded accusations. They want an investigation to look at whether Mr. Comey set up Mr. Trump as president-elect.

Mr. Comey visited Mr. Trump for an intelligence briefing at Trump Tower days before he took office. Mr. Comey broke the news about what he called a “salacious and unverified” dossier now known to have been bought by the Democratic Party from its contractor, former British spy Christopher Steele. It told of Mr. Trump supposedly frolicking with prostitutes in Moscow and overseeing Russia’s hacking of Democratic Party computers.

At the moment Mr. Comey called the scintillating dossier “unverified,” the FBI was using it to justify a wiretap on former Trump campaign volunteer Carter Page.

The briefing instantly leaked to CNN, rocking the transition team with a series of unverified dossier accusations. BuzzFeed then posted the 35-page dossier, which remains unconfirmed publicly.

“There is a question as to whether the FBI included the dossier in the briefing, and possibly leaked that it had done so, in order to provide the media a pretext to report on the dossier,” Mr. Grassley and Mr. Graham wrote.

Democrats, such as Rep. Adam B. Schiff of California, seized on the dossier as authentic and used it to pummel the Trump White House for more than a year.

The Grassley-Graham letter also asserts that information provided to the Judiciary Committee by Mr. Comey in a private interview is directly contradicted by the FBI’s Foreign Intelligence Surveillance Act applications to spy on Mr. Page.

“Did Director Comey intentionally mislead the committee?” the senators say in the letter.

The senators also asked the inspector general to get to the bottom of one key question in the nearly yearlong wiretap warrant.

The FBI asked a FISA judge to approve the surveillance, mostly on the word of Mr. Steele’s dossier. The FBI did not disclose to the judge that the dossier was Democratic Party opposition research paid for by the Hillary Clinton campaign. The Republicans want Mr. Horowitz to find out why there was no disclosure.

“Did anyone express any concerns about the propriety of presenting unverified, uncorroborated claims from the Steele dossier as the basis for a FISA warrant on an American citizen?” the letter stated.

The letter also asked for an investigation into Mr. Steele’s ties to the Russian government and its intelligence services. The dossier’s claims, all of which have been denied by Trump people, were sourced by Mr. Steele to unknown Kremlin operatives. Mr. Steele was once posted at the British Embassy in Moscow while an MI6 spy.

Since the Oct. 21, 2016, FISA warrant allowed the FBI to seize emails and digital messages before the start of the Trump campaign, the senators asked if Obama administration political appointees read those documents.

After the FBI fired Mr. Steele for breaking a pledge not to go to the press, the investigative firm Fusion GPS, his paymaster, set up a back channel to the bureau team investigating the Trump campaign, according to the letter. The flow of information went from Mr. Steele to Fusion GPS to Associate Attorney General Bruce Ohr to FBI investigators.

At the time, Mr. Ohr’s wife, Nellie, worked at Fusion and was investigating Mr. Trump. Mr. Ohr, whose office was steps away from the Obama Justice Department hierarchy, was subsequently demoted.

“Was anyone in the Justice Department, including senior leadership, aware that Mr. Ohr continued to pass information from Steele and Fusion GPS to the FBI even after Steele was suspended, and terminated, as a source?” the letter said.

Mr. Grassley and Mr. Graham already have urged the Justice Department to investigate whether Mr. Steele told a fundamental lie to the FBI that skewed how the bureau filled out its Carter Page wiretap proposals.

At first, their Steele referral to the Justice Department in January was highly classified and redacted. But that changed.

Rep. Devin Nunes, California Republican and chairman of the House Permanent Select Committee on Intelligence, pressed the FBI to turn over FISA material. He then wrote a memo exposing what he considered the abuses and got it cleared by the White House.

Mr. Schiff, the committee’s top Democrat, then wrote a rebuttal, which mostly was cleared by the White House, resulting in more information being declassified.

On the Senate side, that resulted in the lifting of more redactions from the Steele referral.

A key disclosure appeared for the first time in the Grassley-Graham referral, and it came from Mr. Schiff’s rebuttal of a few weeks earlier.

The issue is this: Before a judge, the FBI used a Yahoo News story to bolster Mr. Steele’s claim that Mr. Page, while on a trip to Moscow, visited two high-ranking Kremlin figures and discussed bribes for sanctions. Mr. Page repeatedly has denied this under oath.

The FBI asserted to the judge that Mr. Steele wasn’t the source, even though the Yahoo story continued the exact same information. The public now knows Mr. Steele made that pledge to the FBI.

In a previously redacted sentence taken directly form the FBI’s FISA application, the bureau wrote, “[Mr. Steele] told the FBI that he/she only provided this information to the business associate [Fusion GPS] and the FBI.”

If the FBI is quoting Mr. Steele correctly, then he lied, the senators say.

The following spring, Mr. Steele admitted in a court filing in London that he did in fact speak to Yahoo in September in Washington at the urging of Fusion. It is on that basis that Mr. Nunes says Mr. Steele lied, and that Mr. Grassley and Mr. Graham are seeking an investigation.

Said the referral: “The FBI repeatedly represented to the court that Mr. Steele told the FBI he did not have unauthorized contacts with the press about the dossier prior to October 2016. The FISA applications make these claims specifically in the context of the September 2016 Yahoo News article. But Mr. Steele has admitted — publicly before a court of law — that he did have such contacts with the press at this time, and his former business partner [Fusion founder Glenn] Simpson has confirmed it to the Committee.”

Mr. Comey has defended himself via his Twitter account against corruption charges from Mr. Trump. His memoir is due out April 17, and he has scheduled a busy round of media interviews.

“Mr. President, the American people will hear my story very soon. And they can judge for themselves who is honorable and who is not,” he said.

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