The FBI has never had a female director.
But about 11 people in all are being considered, USA media report.
In a series of tweets, Mr Trump threatened Mr Comey and also warned that he may cancel all White House press briefings. We need to be able to learn from our mistakes, and our successes, so it's in the public interest.
Citizens for Responsibility and Ethics in Washington filed its request for a probe with Deputy Attorney General Rod Rosenstein on Friday. An ex-FBI agent, Rogers drew the backing of the FBI Agents Association, which said his diverse background makes him the best choice.
Sessions has faced questions over whether his involvement in Comey's firing violates his pledge to recuse himself from investigations into Russian interference in the election. He later stated that he had long planned to fire the director, and accused Comey of being a "showboat".
Department of Justice officials have already begun interviewing candidates for the permanent Federal Bureau of Investigation director role.
Fisher was assistant attorney general for the department's criminal division under President George W. Bush. She now works at a law firm.
Before he was sacked on Tuesday, Comey acknowledged the FBI was investigating possible ties between Team Trump and Russian agents seeking to influence the 2016 election.
But it's the court cases and laws stemming from Nixon's secret recordings that could send Trump's tweet backfiring.
"[James Comey] had a lot of pressure put on and he exonerated her", Trump said Friday in his interview with NBC Nightly News' Lester Holt.
Comey's dismissal came on Tuesday, in what the White House alleged was a decision following the advice from the attorney general and deputy attorney general, although Trump later said he was intending to fire Comey "regardless".
During a press briefing, White House spokesman Sean Spicer denied that Trump had asked for Comey's personal loyalty and said the only loyalty required was to the country. Repeatedly, Spicer said he had "nothing further to add on that". Comey has been invited to testify before the Senate Intelligence Committee next week, although he has not yet accepted.
John Conyers and Elijah Cummings' letter noted "it is a crime to intimidate or threaten any potential witness with the intent to influence, delay or prevent their official testimony".
In doing so, though, Trump sought to criticize Comey after suddenly firing him earlier this week.
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