The video shows Saadi Gaddafi screaming as he is beaten by guards on the soles of his feet.
Tripoli’s public prosecutor said he would work to identify the guards shown and take legal action against them.
Campaigners have condemned the footage, which was made available by online site ClearNews.
“No exceptional circumstances justify torture or other ill-treatment,” said Human Rights Watch’s deputy Middle East director, Joe Stork.
Saadi Gaddafi has been held in a prison in the capital Tripoli since last year, after being extradited from Niger on charges of shooting protesters and other crimes committed during his father’s rule.
Another son of Col Gaddafi, Saif al-Islam, was last week sentenced to death in absentia over crimes committed during the 2011 uprising that eventually toppled the former Libyan leader.
He is being held by a former rebel group from the town of Zintan that refuses to hand him over.
Libya has been in chaos since the overthrow of Col Gaddafi, with two factions claiming to control the country.
The internationally-recognised parliament is based in Bayda and Tobruk in the east, while the capital Tripoli in the west is held by rivals Libya Dawn.