Tiernan-Locke, who was suspended from competition for two years up to December 2015 on anti-doping grounds, made the allegation during an interview with the BBC, aired on Thursday evening.
He also commented Bradley Wiggins's and Team Sky's use of triamcinolone injections, which came to light when Wiggins's therapeutic use exemption (TUE) certificates were published online by hackers.
"There was a time I rode the World Championships and we were offered a painkiller called tramadol," said the 31-year-old from Cornwall.
"I wasn't in any pain so I didn't need to take it, and that was offered freely around. It just didn't sit well with me at the time. I thought, 'I'm not in any pain', why would I want a painkiller?'"
Tramadol is a controversial drug within cycling. Although it is not currently on the World Anti-Doping Agency's Prohibited List and is not banned, it is on WADA's 'watch list' as being a substance that could potentially be abused by athletes. Its side-effects can cause lapses in concentration, with the potential for causing crashes.
Commenting on Wiggins's use of anti-allergy drug triamcinolone before the 2011 and 2012 Tour de France and 2013 Giro d'Italia, Tiernan-Locke told the BBC: "I don't want to insinuate anything, but the timing doesn't look great. You assume if you had a need for such a thing it would be consistent throughout his career, that you'd use it year in year out, so from that point of view it looks suspicious.
"But from the other perspective, we've got a guy who's favourite for the general classification in these big races, so for a team like Sky who are so thorough, they don't want to leave anything to chance so why risk these allergies derailing their best-laid plans, so I understand it."
However, Tiernan-Locke said that he thinks the episode surrounding Wiggins's TUEs may have 'tainted' Team Sky's image.
"People I have spoken to, out of training, it has kind of tainted their image somewhat," he said. "I think their modus operandi was to put things out in the public domain and look transparent and not have anything to hide. It's somewhat less than transparent."
TUEs are produced when an athlete is prescribed a drug on WADA's Prohibited List for a legitimate medical reason. TUEs are not evidence of wrongdoing.
Wiggins told the BBC in an interview at the end of September: "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."
Tiernan-Locke was suspended for competition for two years due to irregularities in his anti-doping biological passport, which is used to detect the use of banned performance-enhancing substances or blood doping.
The irregularities relate to his time with the Endura team, directly prior to being signed to Team Sky for the 2013 season. Team Sky terminated Tiernan-Locke's contract when he was suspended.
Tiernan-Locke spoke about his own doping sanction, and maintains that he did no wrong. He returned to racing in 2016 after his ban expired in December.
"I'd put so much hope in the result that I'd get cleared and be racing again, so that that was a bitter pill to swallow," he said. "But time has kind of healed that. I didn't touch a bike for ages, I thought I was done with it. Now I'm trying to put something back into the sport again."
British Cycling has been contacted for comment.
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