McQuaid said Armstrong should travel to UCI headquarters in Switzerland to tell all about his doping history and offer to help clean up the sport.
"He should jump on his private plane, come to Switzerland and say, 'What should I do?'" McQuaid told reporters. "He still hasn't apologized to the sport of cycling. If he has information that is valuable to the sport he should come forward."
McQuaid said Armstrong should also meet with the U.S. Anti-Doping Agency and World Anti-Doping Agency to provide full details on how he cheated to win the Tour de France seven times. The American was stripped of all the titles last year after admitting to doping.
"Everyone accepts he has not come clean," McQuaid said. "He should sit down and work with us."
McQuaid and the UCI have come under fire for failing to catch Armstrong and the federation has been accused of complicity and coverups in his cheating.
But, in a wide-ranging interview with reporters on the sideline of the SportAccord Convention, McQuaid repeatedly defended himself, the UCI and former president Hein Verbruggen.
"I do not think the UCI made mistakes," the Irish official said. "The facts show the UCI was always the most advanced federation in the fight against doping. The problem was the products that couldn't be tested for at the time. There were no tests available for the products. The UCI was not to blame.
"Ten or 15 years ago, the armoury was much weaker. Today we are spending 7.5 million euros ($9.6 million) a year on testing. We are not spending 7.5 million euros ($9.6 million) to let cheats get away."
McQuaid said the UCI tested Armstrong 200 times between 1999 and 2005, while USADA tested him 12 times during that period.
"All the blame has been put on the UCI," he said. "Maybe USADA and WADA should also take some responsibility."
McQuaid said he has invited USADA chief executive Travis Tygart for talks in Switzerland. Tygart, who has been one of the UCI's harshest critics, has accepted and a date for the meeting is being arranged, McQuaid said.
The UCI head claimed he never suspected Armstrong was cheating during his dominance of the Tour de France, saying he had been inspired by the rider's comeback from cancer.
"I didn't know what he was up to. I was fooled," McQuaid said. "I believed there was no way a man so close to death would go and start putting stuff into his body that could be dangerous. ... I gave him the benefit of the doubt."
McQuaid, who has faced calls to step down in the wake of the Armstrong revelations, said he has never considered resigning.
"I firmly believe I am making a difference," he said. "I firmly believe there is a change in the peloton and the sport has changed."
McQuaid, who has headed the UCI for eight years, is up for re-election for a third term in September.
"I want to eradicate doping," he said. "I want to see this through. I want to finish what I started."
Despite the continuing questions over the UCI's role and the fallout from the Armstrong case, McQuaid believes the scandal will fade away and the sport has a rosy future.
"I believe it's in the past,' he said. "I believe it will pass over. It will go away. The sport has a bright future. We're becoming truly global."
McQuaid cited the growth of the sport in Asia and Africa, and predicted a "black African will make the podium of a major tour within the next six years."
On another subject, he said the UCI is proposing the addition of BMX freestyle and mountain bike elimination events for the 2016 Olympics in Rio de Janeiro. Also still in discussions, he said, is the possibility of bringing skateboarding into the Olympics under the UCI umbrella.