The cause of his death was not immediately clear. CBS News, the BBC US partner, confirmed his death.
"With deep sadness, we are sharing the news that Bob passed away" on Friday night, his family told the AP in a statement. "His family asks that their privacy be respected."
Mueller is survived by his wife of nearly 60 years, Ann Cabell Standish, their two daughters, and three grandchildren.
Mueller's special counsel inquiry put Donald Trump's 2016 campaign under a microscope, drawing harsh criticism from the US president.
The president wrote on Truth Social on Saturday: "I'm glad he's dead. He can no longer hurt innocent people!"
Mueller's former employers and colleagues praised him as a longtime public servant. Both of the presidents he served under as FBI director - George W Bush and Barack Obama - paid tribute.
Bush, who appointed Mueller to lead the FBI, said he was "deeply saddened" by his death.
"In 2001, only one week into the job as the sixth director of the FBI, Bob transitioned the agency mission to protecting the homeland after September 11," he said. "He led the agency effectively, helping prevent another terrorist attack on US soil."
Obama called him "one of the finest directors in the history of the FBI" and commended his "relentless commitment to the rule of law and his unwavering belief in our bedrock values".
by Kayla Epsteinand and Anthony Zurcher, BBC | Read more:
Image: Getty Images
[ed. Mr. Mueller was true patriot. Can you imagine the task of transforming the FBI in the wake of 9-11? True to form, the most classless president in American history had to weigh in with a statement of historic classlessness. Maybe he was just channeling his own eventual eulogy.]
