Flight Lieutenant Kirsty Stewart, the first woman pilot in the aerobatic display team, and then-Squadron Leader Ben Murphy were interviewed to see if any rules had been broken when claims of a relationship between them emerged.
A source for the RAF said no action was taken following the 2011 inquiry.
"The two individuals were interviewed by their command chain," he said. "At that point, and the interviews given, it led to there being no action taken."
Asked if the RAF had a view on a report that Flt Lt Stewart and Sqn Ldr Murphy were engaged to be married, he said the RAF did not discuss personal matters."
Both fliers have now left the elite Red Arrows team.
Flt Lt Stewart, 36, joined the team for the 2010 season, having seen off competition from up to 40 other pilots. She is now a squadron leader, having moved to a ground-based role.
Mr Murphy, 38, now works for Lockheed Martin.