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.
"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.