The video, which was posted online on Thursday by Mekameleen, an opposition TV channel based in Turkey, appears to show two blindfolded men being shot in the head by uniformed soldiers.
In the video, one young blindfolded man is aggressively questioned by the officers who ask him if he is a member of the Abu Sanana family.
When he responds "No" and says he is from al-Awabadah, they ask him where his grandfather lives before forcing him to the ground and removing his blindfold.
One of the soldiers then shoots him in the head.
A person off-screen can be heard repeatedly shouting "Not just the head" as a second soldier then fires at the body on the ground.
Al Jazeera could not independently verify the video's authenticity. The Egyptian government has barred journalists from reporting on events in the Sinai Peninsula, where the army is battling a local affiliate of the Islamic State of Iraq and the Levant ( ISIL) group
Possible war crimes
The leaked video appears to depict part of a raid that the Egyptian army posted on social media on December 6.
That post included photos of three men in a field with guns next to them, who had been "killed in a raid" on an explosives storehouse.
Ahmed Mefreh, a lawyer and human-rights researcher, told Al Jazeera the killings amounted to war crimes.
"In the video, we see executions of unarmed people by forces wearing military uniform," he said.
"They are being shot in a direct way. International law considers this a war crime that clearly has to be investigated."
Yehia Ghanem, Al Jazeera's Middle East analyst, said the names, villages and tribes mentioned by the victims lent authenticity to the video.
"It's important that the military call for an independent and impartial investigation in order to clear their name and reputation," he said.
"The political leadership needs to take the reigns of the military. For the past few years they've been acting like a runaway train. Actions such as this endanger social peace in the region and could accelerate growing calls for Sinai to separate."
Last month, Human Rights Watch (HRW) accused Egyptian forces of executing at least four men in January 2017.
It said the killings appear to fit a pattern of abuse against civilians by both military and internal security forces who have been deployed to Sinai.
On Friday, HRW said in a statement that other videos and photographs it had reviewed corroborated the incidents shown in the latest leaked video from Sinai.
"These outrageous killings confirm that Egypt's counterterrorism campaign in the Sinai is out of control," said Joe Stork from the group's Middle East division.
"Egypt's allies cannot claim ignorance about these deadly abuses."
Egypt has for years been battling an armed campaign in the thinly populated Sinai Peninsula.
The offensive has gained pace since the military overthrew democratically elected president Mohamed Morsi of the Muslim Brotherhood in mid-2013.