Cookson, president of British Cycling, is so far the only person to challenge Irishman McQuaid, who is seeking a third term in the post.
The 62-year-old announced his key election points on Monday, which included the establishment of an independent anti-doping unit, only for McQuaid to denounce the manifesto as "half-baked, fundamentally flawed and financially impractical".
In a statement, Cookson said: "On Monday I set out a new agenda for the UCI and cycling which has already received very strong support from around the world.
"I have been truly encouraged by the messages I have received following the launch and the serious and considered way which members of the cycling family and the media have responded to the direction I want to set.
"The response from Pat McQuaid to my manifesto has once again demonstrated exactly why restoring credibility to the UCI and cycling in general was the number one recommendation of the recent Deloitte consultation with the sport's stakeholders," he added.
"As we enter the next stage of the presidential election, it is clear that the choice that has to be made is between two different approaches to the work of the UCI and two different visions for our sport."
Cookson has based his candidacy on restoring trust and credibility in the UCI as the organisation struggles to deal with the aftermath of the Lance Armstrong doping scandal and allegations it did not do enough to catch the American, who was stripped of his seven Tour de France titles last year.
- Pat McQuaid