Wiggins was permitted to take the TUE just days before the 2012 Tour de France, which he won, as well as the 2011 Tour and the 2013 Giro d'Italia.
“It was prescribed for allergies and respiratory problems,” he said. “I’ve been a lifelong sufferer of asthma and I went to my team doctor at the time and we went in turn to a specialist to see if there’s anything else we could do to cure these problems,” said Wiggins.
“And he in turn said: ‘Yeah, there’s something you can do but you’re going to need authorisation from cycling’s governing body [the UCI].’”
The 36-year-old, who will retire at the end of the season, said that he needed evidence from a specialist that was then scrutinised by three independent doctors as part of the process.
“This was to cure a medical condition. This wasn’t about trying to find a way to gain an unfair advantage, this was about putting myself back on a level playing field in order to compete at the highest level.”
Prentice Steffen was multiple Olympic champion Wiggins's doctor at Garmin Slipstream, with whom he finished fourth in the 2009 Tour de France before the Briton joined Team Sky.
Wiggins's TUE history was made public last week when his medical records were leaked, along those of several major athletes.
Steffen told the BBC's Newsnight programme on Friday the leaked details of Wiggins's TUEs did not "look good".
"I was surprised to see there were TUEs documented for intramuscular triamcinolone just before three major events -- two Tours de France and one Tour d'Italia," Steffen said.
"You do have to think it is kind of coincidental that a big dose of intramuscular long-acting corticosteroids would be needed at that ... exact time before the most important race of the season.
"I would say certainly now in retrospect it doesn't look good, it doesn't look right from a health or sporting perspective."
Wiggins' TUEs, exemptions which allow athletes to take substances banned by the World Anti-Doping Agency if they have a medical need, were released by Russian hackers. A group known as “The Fancy Bears” have leaked the private medical records of dozens of athletes after gaining access to the WADA database.
There is no suggestion any athlete has broken anti-doping rules, but Wiggins, the five-time Olympic champion, has found himself the focus of the most scrutiny.
American tennis stars Serena and Venus Williams, American gymnast Simone Biles and Wiggins's Team Sky team-mate Chris Froome have also been the subject of leaks.