The country's top judicial regulatory body ordered the probe after pictures widely circulated on social media showed the two apparently asleep in the trial of a policeman accused of killing a protester during the mass street demonstrations in June.
The Higher Board of Judges and Prosecutors has requested video recordings of Monday's hearing in the main Ankara court to decide whether further action is needed, local media reports said.
The incident was widely mocked on Twitter, with hundreds of users sharing a picture of graffiti that read "Police shot, judge slept, don't be silent".
At Monday's hearing - where the accused policeman appeared in disguise and via videolink - the court withdrew from the case and referred it to a higher tribunal because of concerns voiced by the victim's family about the tribunal's impartiality.