The American cyclist was stripped of his seven Tour de France titles and banned for life last year for doping.
Armstrong subsequently admitted in a television interview last month that he had taken performance-enhancing drugs in each of his Tour wins.
The UCI's credibility took a further blow when it disbanded a three-person independent commission set up to investigate whether or not the world governing body had helped Armstrong to conceal his drug-taking without sending it a single document.
It said instead it wanted to be part of a truth and reconciliation commission.
At a World Anti-Doping Agency media symposium in London on Tuesday, WADA president John Fahey said he had received a letter late on Monday from the UCI again requesting a truth and reconciliation commission.
"Was it asking?," he said. "It was more telling. It nominated four people from the organisation which did not include (UCI president) Pat McQuaid who would be representing UCI in such discussions."
Fahey said WADA had said all along it was more than happy to work out a way to deal "with this constant crisis which seem to surface in cycling".
He said, though, the process had to be under the management and control of the original independent commission.
"So I put those terms back, that's the starting point, it hasn't changed for several weeks. If they are serious they will talk to us," he said.
Fahey said the terms truth and reconciliation were "wonderful fancy words".
"I think we leave that where we should leave it, where it had some meaning and that is in days gone by in South Africa," he said.
"Only cycling can heal the problems cycling has, they're independent, they run their own sport, the same as any other sport in the world.
"If the members are prepared to continue to allow this lurching from one crisis to another then I guess we are going to continue to read about turmoil in that sport for some time yet.
"I would hope that within the root and file members of cycling there is recognition that it can't continue this way without there being some dire consequences down the track."
- Sports & Recreation