The pilots on board both Boeing 747s were warned to take immediate evasive action as they closed on each other 34,000ft above the Scottish countryside.
But they either “misheard or misinterpreted” the instructions of air traffic controllers and instead flew closer.
The report by Airprox, a flight safety watchdog, found that the planes came within 100ft vertically, and less than three nautical miles horizontally, of a collision before the emergency ended on 23 June.
In its findings, Airprox said: “It was possible that the crews may have been distracted because this would have been about the time that they would have been receiving their Oceanic clearances on data-link.
“Another possibility mooted by an airline-pilot member was that, having settled into their trans-Atlantic routine, it was unusual for pilots to be issued with avoiding action instructions at that altitude and location.
“Expecting only routine information to be transmitted at that time, they may have been perplexed by the avoiding action information and instinctively responded without properly assimilating it.”
A collision was avoided when both pilots diverted, with one climbing and the other diving.
When the alarm was first sounded the jets were about ten miles apart, but within a minute they were less than three miles apart horizontally.
According to the report, the near miss happened as the two jets, with around 1,000 people on board, were about 30 miles north of Glasgow and preparing to cross the Atlantic.
An air traffic controller noticed that the two planes “were coming into conflict with each other”. The controller instructed the crew of one Boeing to turn left and the other to turn right.
The report states: “He observed that the B747(1) did not appear to be taking the turn. The pilot of the B747(2) queried whether he had been given a turn and he reissued it.”
Within a minute the two aircraft were 6.6 nautical miles apart and still on converging headings.
Twenty seconds later, with the two planes less than five miles apart, the Scottish air traffic controller instructed one of the two pilots to “descend now immediately”. Airprox has assessed the incident as a Category C emergency in which there was no risk of actual collision.
The report states: “As to the risk, because B747(1) had B747(2) in sight as it was turning towards it, and both aircraft reacted, thereby establishing standard vertical separation by a horizontal distance of 2.8 nautical miles, the board opined that there was no risk of a collision.
A second Airprox report has revealed the pilot of an American fighter jet activated his plane’s weapons radar system in an attempt to avoid a possible collision with a Scottish passenger plane on a flight to Humberside near Leconfield, on 16 May.
The F-15E jet passed within 1,200ft of a Jetstream S41 jet flying to Humberside from Aberdeen with 31 passengers and crew on board.
The near-miss has been identified as a Class B incident by the Airprox, meaning the safety of the aircraft was compromised.
The report states: “The incident occurred when the F15 crew commenced a rapid climb from low level into proximity with the aircraft.
“The F15E pilot climbed into conflict with the JS41, which he did not see.
“The F-15E pilot said subsequently that he detected a slight hesitation or inflection in the controller’s voice and immediately switched the radar from its ‘search’ mode to ‘guns’ mode, a move which still failed to reveal the JS41, probably because it was already behind him.”