A Republican congressman is calling on Special Counsel Robert Mueller to resign from his role leading the Russia investigation, citing the investigator’s friendship with former FBI Director James B. Comey.

Rep. Trent Franks said Mr. Mueller’s relationship with Mr. Comey should make him ineligible to lead a probe of Russian interference in the presidential election and any possible coordination with the Trump campaign — the investigation includes whether the president fired the FBI director in an effort to undermine the investigation.

“Those who worked under them have attested he and Jim Comey possess a close friendship, and they have delivered on-the-record statements effusing praise of one another,” Mr. Franks, Arizona Republican, said in a statement issued Tuesday. “No one knows Mr. Mueller’s true intentions, but neither can anyone dispute that he now clearly appears to be a partisan arbiter of justice.”

Mr. Franks, a member of the House Judiciary Committee, cited federal law that prevents the special counsel from serving in the role if the person has a conflict of interest. In Mr. Mueller’s case, the lawmaker said that conflict is “a personal relationship with any person substantially involved in the conduct that is the subject of the investigation.”

Mr. Mueller was appointed to lead the FBI by President George W. Bush. He headed the bureau for 12 years before Mr. Comey was appointed in 2013, but the two men worked together for a time when Mr. Comey served as deputy attorney general.

The decision to appoint Mr. Mueller as special counsel fell to Deputy Attorney General Rod Rosenstein as Attorney General Jeff Sessions recused himself from all investigations related to the presidential campaign. The appointment was made a week after Mr. Trump fired Mr. Comey.

Mr. Franks’ call for Mr. Mueller’s resignation comes as other lawmakers have expressed concern that the president may be looking to fire his attorney general as a vehicle for removing the special counsel or curtailing his investigation.

Mr. Trump has expressed disappointment over the fact Mr. Sessions recused himself from the investigation, meanwhile members of his administration have been critical of some of the attorneys Mr. Mueller has selected for his special counsel team and the widening scope of the investigation.

At least five members of the team have given donations to Democratic candidates. Mr. Mueller is a registered Republican.