James Dean Blevins is accused of threatening to kill Flake because he was "tired" of the Republican "interrupting" President Donald Trump.
A grand jury in Arizona indicted Blevins.
A federal magistrate in Chicago said in an order late Wednesday, that prosecutors in Arizona didn't agree to transfer the case to Chicago.
The Chicago Sun-Times reported Blevins left a voice message at Flake's office, saying, "I am tired of him interrupting our president, and I am coming down there to take him and his family out."
Blevins' lawyer said his client had been drinking.
The magistrate released Blevins to a family member’s custody until his next court date in Arizona.