The US-based Institute for Justice and Democracy in Haiti said it is now representing 8,000 victims and families of the more than 8,100 dead from the epidemic that erupted in October 2010.
IJDH director Brian Concannon said the suit would be launched in New York if the United Nations did not respond by the deadline, and could also be taken in Europe.
The United Nations said in February that it was legally immune from action over the epidemic that has sickened more than 650,000 people.
The epidemic has been sourced to a river that runs next to a peacekeepers' camp in the central town of Mirebalais, where Nepalese troops had been based. The strain of cholera is the same as one endemic in Nepal.
Institute lawyers say they are seeking $100,000 for the family of each victim who died from cholera and $50,000 for each survivor. That could take the potential claim into several billion dollars.
"We are looking for an urgent and adequate response from the UN," Concannon said. "Immunity cannot mean impunity."
The cholera epidemic dealt a new blow to Haiti as it struggled to overcome the strife caused by a January 2010 earthquake that killed 250,000 people.
The UN said in February that the complaint from the victims' lawyers was "non-receivable" under a 1946 convention setting out the UN's immunities for its actions.
UN Secretary-General Ban Ki-moon launched a $2.2 billion appeal in December, however, to raise money to provide clean water and health facilities in the deeply impoverished Caribbean nation.
Lawyers sent their formal response to the UN on Tuesday.