A federal judge has begun interviewing candidates to serve as a special monitor to oversee the FDNY, but parties involved in the litigation have yet to agree on the choice.

Brooklyn federal Judge Nicholas Garaufis said at a hearing yesterday that he has been holding private discussions with “a very impressive group of candidates,” but lamented that city officials, the Justice Department and a fraternal organization for black firefighters have not found common ground.

“I think it’s troubling that you could not all embrace at least one candidate,” Garaufis said at a hearing.

Among the dozen people currently under consideration for the position are former judges, former prosecutors, litigators, arbitrators, mediators, former city officials, law professors and law-school deans, said Richard Levy, a lawyer representing the Vulcan Society, a fraternal group for black firefighters, after the hearing.

Three weeks ago, the judge ruled that broader judicial oversight of the FDNY is need because of the city’s “pattern and practice of discrimination against black firefighter candidates,” as well as minorities already working for it.