Hegseth, speaking to TV outlet Real America's Voice News, said the execution of Hasan will take place at Fort Hood on December 3.
Hasan was convicted of killing 13 people and injuring 32 others in a shooting at the military base in 2009.
"We'll make sure that people are able to watch it, that it's public," Hegseth said.
When asked if Hasan had made any final requests or reached out to the Trump administration, Hegseth simply replied: "Don't care. We'll see him on December 3rd."
Officials in the Pentagon confirmed on Thursday the execution will be livestreamed and that more details would follow.
Pentagon spokesperson Sean Parnell had said Monday that President Donald Trump had approved the execution of Hasan.
The White House on Thursday immediately referred questions about the plans for the livestreamed execution to the Pentagon.
Hasan, an Army psychiatrist who held the rank of major, walked into a medical building where soldiers were were waiting in long lines to receive immunisations and medical clearance for deployment. He shouted "Allahu akbar" - Arabic for "God is great!" - and opened fire with a laser-sighted handgun.
During the trial it was revealed that Hasan spent weeks planning the deadly attack, and his preparation included videotaping a sales clerk showing him how to change the magazine on the pistol that he purchased.
He would later go to a gun range outside Austin and ask for advice on how to reload with speed and precision from an instructor.
Soldiers later testified that Hasan's rapid reloading made it all but impossible to stop the shooting. Investigators recovered 146 shell casings inside the building and dozens more outside, where Hasan shot at the backs of soldiers fleeing toward the parking lot.
Hasan said it was an effort to stop them from going to Afghanistan and killing his fellow Muslims. A military judge barred him from presenting that as a defence during his trial.
Hasan made almost no effort to defend himself in court, telling jurors when he acted as his own counsel that evidence would show he was the shooter and describing himself as a soldier who had "switched sides."
Hegseth said the US military hasn't executed a service member by firing squad since World War II. The last military execution took place on April 13, 1961, when Army Private John Bennett was executed by hanging.
