The Justice Department signaled Wednesday in a letter to Republican Sen. Lindsey Graham that Deputy Attorney General Rod Rosenstein will not recuse himself from Special Counsel Robert Mueller's Russia investigation unless department officials find it necessary.

Assistant Attorney General Stephen Boyd wrote that Rosenstein "would recuse from all or part of any matter if recusal were deemed appropriate by Department officials who are informed about the matter.

"Otherwise," Boyd continued, "[Rosenstein] has a responsibility to fulfill his oath to well and faithfully execute the duties of his office."

Graham sent a letter to Rosenstein last week asking the deputy attorney general whether he considered himself a "potential witness" in the Mueller probe due to his role in the firing of FBI Director James Comey. President Trump cited a memo authored by Rosenstein to justify Comey's dismissal in May 2017.

Fox News reported last week that Rosenstein claimed to since-ousted FBI official Andrew McCabe that Trump had asked Rosenstein to reference Russia in his recommendation to fire Comey. Rosenstein declined, and instead focused on Comey's handling of the Hillary Clinton email probe.

A source with knowledge of the investigation told Fox News that Trump just wanted Rosenstein to mention that Comey told him he was not under investigation in the Russia case. Trump himself, though, also has cited the Russia probe in explaining his decision to fire Comey.

Rosenstein appointed Mueller to oversee the Russia investigation after the recusal of Attorney General Jeff Sessions, a move that repeatedly has drawn Trump's ire.

Fox News' Catherine Herridge and Judson Berger contributed to this report.