McQuaid, bidding for a third mandate as president of the International Cycling Union, angrily hit out at suggestions that he had broken the rules governing the nomination of candidates, describing the claim as "outrageous.”
Cookson, head of British Cycling and the only challenger so far, replied that McQuaid's remarks were "another attempt to denigrate the current presidential process.”
Irishman McQuaid, who has been president since 2005, failed to win the backing of his own country's Irish Cycling federation but received nominations from the federations of Switzerland, Thailand and Morocco.
However, he later received a letter from British Cycling's lawyers which suggested the UCI had accepted his nominations from the Thai and Moroccan federations after the closing date, McQuaid said.
"That is an outrageous suggestion," said McQuaid in a statement.
"Brian must immediately make a statement on whether he believes that to be true and if he believes otherwise he has duty to ensure that this allegation is publicly withdrawn.”
McQuaid confirmed that the Swiss, Thai and Moroccan nominations were all received before the deadline in accordance with the UCI rules.
"As the president of British Cycling, Brian Cookson must explain his decision to allow his federation - that is funding his campaign - to behave in this way and to use its considerable financial clout to employ lawyers to challenge issues in the election," said McQuaid.
"I do not fear an open election and I am not at all concerned by my ability to secure the support and votes that I require to be re-elected as UCI President," he added.
"While it would appear that Brian has lost confidence in his own ability, I continue to challenge him to allow the UCI Congress and its voting delegates to decide.”
Cookson, who has been president of British Cycling since 1997, quickly hit back with a statement of his own.
"Sadly today we have seen yet another attempt by the existing UCI President, Pat McQuaid, to denigrate the current presidential election process," he said.
"I want nothing more than an open and properly conducted democratic election and vote for the UCI presidency. To suggest otherwise is nonsense.
"It is also true that I, and many in our sport, have legitimate and growing concerns about the retrospective rule bending and attempted manipulation that is taking place at present.
"In my view it is therefore absolutely correct that British Cycling and others have raised concerns regarding proposed rule changes which have a direct impact on the election process now under way. These concerns need to be addressed.
"Far from ducking these issues, for the good of cycling and the reputation of the UCI, it is critical that openness and transparency guide our procedures and not desperate manoeuvres and outbursts by Mr McQuaid.”
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.
McQuaid has already described Cookson's election manifesto as "half-baked, fundamentally flawed and financially impractical". The election is scheduled to take place in September.