Arutyunian, 27, faces related charges in Georgia, where he is in custody.
A Justice Department official said the U.S. government will wait until Arutyunian is tried in Georgia before deciding whether to seek his extradition to face trial in the United States.
The charges stem from an incident on 10 May while Bush was giving a speech in Tbilisi's Freedom Square. Arutyunian is accused of throwing a hand grenade into the crowd about 30 meters from the stage where Bush was standing. The live grenade failed to detonate.