Mueller faces a mandatory 10-year retirement in September.
Fitzgerald has been at the helm of a number of high-profile cases, including the investigation into the leak of former CIA officer Valerie Plame Wilson's identity and alleged corruption by former Gov. Rod Blagojevich.
Other names being floated, according to the reports, include New York City Police Commissioner Raymond Kelly and Transportation Security Administration chief John Pistole.
The leading candidate is reportedly another Chicago native, Michael Mason. Mason is a former top official at the FBI's Washington field office currently serving as Verizon Communications' top security chief.
President Barack Obama should name Mueller's replacement soon.