The documentary, Bin Laden, the failings of a manhunt, was made by journalists Emmanuel Razavi and Eric de Lavarene, who have worked for several French media outlets in Afghanistan.
Razavi said the soldier told them it took roughly two hours for the request to reach the US officers who could authorise it, but the anonymous man is also quoted in the documentary as saying: "There was a hesitation in command."
Razavi said several sources told them the sightings were six months apart and they declined to be more specific.
Christophe Prazuck, a French armed forces spokesman, said "that never happened" when asked about the bin Laden sightings.
Bin Laden, who planned the September 11 attacks which killed nearly 3,000 people, is believed to be hiding in the mountains along the Afghanistan-Pakistan border.
France has roughly 200 elite troops operating under US command near the border. Paris announced on Sunday that it was withdrawing them in the new year.
France is part of the 32,000-strong Nato-led International Security Assistance Force (ISAF) in Afghanistan, which took command of the war against the Taliban from US-led forces in October and has since carried out a series of offensives.
ISAF's special forces were deployed in 2003 to bolster Operation Enduring Freedom, a US-led campaign against the Taliban and al-Qaeda in response to the September 11 attacks.
Afghans questioned in the documentary said they believed that the US was not interested in finding bin Laden, despite the $25m price Washington has placed on his head.
The documentary stopped short of that conclusion but raised questions about the US hunt for bin Laden, such as whether Washington is more concerned about preserving stability in Pakistan, where many support bin Laden, than in finding him.
In September, George Bush, the US president, dismissed as an "urban myth" the idea his administration had become distracted from its effort to track down bin Laden.