According to the report, the British vessel HMS Vanguard and the French U-boat Triomphant were both submerged at the time and on separate missions in the mid-Atlantic at the time of the incident, on February 3 or 4.
The daily cited a senior British Navy source as saying that, while it was unlikely that the collision could have sparked a nuclear explosion, the potential consequences were "unthinkable."
"A radiaoctive leak was a possibility. Worse, we could have lost the crew and warheads. That would have been a national disaster," the source said.
Both vessels were outfitted with ultra-modern sonar equipment, but also with anti-sonar technology that may have prevented the boats from detecting each other.
Both submarines were damaged in the incident but there was no damage to either the nuclear propulsion units or the nuclear-armed missiles the boats were carrying, the report said.
The Vanguard, which has a crew of 101, is one of four British V-class submarines, each of which is armed with 16 ballistic missiles.
The online edition of the French weekly Le Point quoted the Defence Ministry in Paris saying that the Triomphant, with a crew of 140, had collided with a "submerged object" and that the accident had damaged the ship's sonar dome.
There were no injuries reported among the French crew, Le Point said.
The British Defence Ministry refused to comment on the incident, other than to say that it had not diminished the country's nuclear defence capacity.
Both Britain and France have opened inquiries into the accident.