The International Cycling Union said late Monday it is preparing an amendment to its statutes, which could safeguard McQuaid's nomination for the September election.
Cookson denounced the tactic on Tuesday as "a clear sign of desperation" by his opponent, who must also give up his International Olympic Committee membership if the presidency is lost.
McQuaid is trying to secure a valid candidacy after his home Irish federation withdrew its support and a nomination from Switzerland, where he lives, is being legally challenged next month.
Malaysian officials now propose changing rules which would allow any two UCI members worldwide, not just home federations, to nominate a candidate — despite the original deadline passing in June.
"The Malaysian Federation and (Asian confederation) state that their aims are to reinforce the independence of future UCI Presidents by ensuring they are able to carry out the role based on serving the global interests of cycling, independently from those of any single nominating national federation," the UCI said in a statement.
The proposed rule change was met with opposition from USA Cycling.
"At this critical time for our sport, we must all stand together and demand strict adherence to the principles of integrity, fair play, transparency, ethical conduct, and good governance," said Steve Johnson, USA Cycling president and CEO. "A dramatic midstream change to the procedures governing the ongoing election is inconsistent with these principles and no more sensible than changing the rules of a bike race after the race has started."
Mike Plant, the U.S. delegate on the UCI board, also wrote to the governing body protesting the change as "unconscionable, unethical, dishonest, unprofessional, manipulative and destructive."
McQuaid, who insisted in a statement Tuesday that "no one has broken the rules" makes his bid for a third four-year term against a background of widespread attacks on the UCI and its credibility. Those intensified in fallout from the Lance Armstrong doping affair and continued revelations of an endemic culture of doping while the UCI was led by McQuaid's predecessor and mentor, Hein Verbruggen.
The UCI has pledged to create an independent panel to investigate claims it colluded in protecting Armstrong from scrutiny during his career, and that $125,000 donated by the now-disgraced American rider was paid to cover up suspicious doping tests.
Cookson was nominated by British Cycling, which he has led for 17 years. He promises to restore cycling's reputation and create an independent body running the sport's anti-doping program.
"It is no wonder that many in the cycling family as well as fans and sponsors have lost faith in the UCI to govern ethically when the man at the top of the organization is prepared to embarrass an entire sport in an attempt to try and cling onto power," he said in a statement.
"What sort of organization attempts to rewrite the rules once an election has actually begun — it smacks of attempted dictatorship," said Cookson, a member of the UCI management board since 2009.
With McQuaid needing a legitimate nomination to stand for election, he now has support from Thailand and Morocco, where he is a member of their national cycling bodies, the UCI said.
"I respect that his (Cookson's) horizons however do not stretch much further than British Cycling," said McQuaid, who has staked much of his presidency on globalizing the sport. He added that the Asian proposal was first suggested to UCI board members in May.
"It has now been formally submitted in accordance with the rules and it is for congress to decide whether it should be accepted or rejected," McQuaid said.
The amendment can be voted on at the Sept. 27 election meeting in Florence, Italy, and retrospectively apply an August deadline for nominations.
In Florence, a 42-voter electoral college will then choose the president by secret ballot.
Cookson's campaign office said his supporter Mike Plant, the United States delegate on the UCI board, also wrote to the governing body protesting the proposed rules change as "unconscionable, unethical, dishonest, unprofessional, manipulative and destructive."
McQuaid's Swiss nomination is set to be judged by a national federation tribunal in Zurich on Aug. 22.
Three members of Swiss Cycling have challenged an endorsement of McQuaid's election bid, arguing it is "tainted on both procedural and substantial grounds."
AP Sports Writer Dave Skretta contributed to this report.