It follows a catalog of perceived errors by the UCI, including the acceptance of a $100,000 donation from Armstrong in 2002 -- four years before McQuaid took up his role.
The money was used by the UCI to buy a Sysmex machine, a contraption which is used to analyze blood samples.
It is a decision which McQuaid concedes should not have been made.
"I would say and we have said, we have admitted that in hindsight -- and of course hindsight is an exact science -- that it would have been better had we not accepted those," he said.
"But we took them at that time in good faith and we used them for the uses that we said we had put them to."
Cookson has been heavily critical of the way McQuaid has led the UCI and has pledged to restore faith in the organization.
He has promised to create a completely new independent anti-doping unit, which would work alongside the World Anti-Doping Agency (WADA).
McQuaid, who is aiming to secure a third term in office, remains philosophical about the future and the battle against doping.
"Nobody is silly enough to say that you're 100% confident that the peleton is 100% clean," added the UCI president.
"That's just not possible. It's not going to be possible in any sport or in any parts of society, but certainly I think the large majority of cyclists are now going into their careers, not wanting to get involved in doping.
"And there's evidence of many riders who come from strong teams, with strong anti-doping backgrounds all winning races and when they get across, you know, when they're doing their press conferences, they are saying I'm an example of how you can win a race clean
"I don't think it'll ever be beaten because in every part of society there are cheats.
"There are people who look for a short-cut, who try to win by devious means or whatever.
"Whether it's cheating by taking drugs or cheating in any other fashion, there are always going to be people like that in society and in sport ."
There is one avenue which remains open to McQuaid though -- a route which involves Armstrong.
The opportunity to talk to the disgraced cyclist in a bid to improve doping control remains a possibility -- and McQuaid is refusing to rule out such a move.
"He certainly has more knowledge," he said of Armstrong.
"If he was prepared, in terms of coming to the UCI, not necessarily to me and explain in more detail or give us more assistance, give us more information and try to help us in planning the future on the fight against doping, we'd certainly be interested in speaking to him."