The campaign, led by the United States and Costa Rica and backed by some predominantly Catholic nations, seeks to overturn a recent vote in the assembly's legal committee to defer consideration of the treaty until 2005.
The panel, which includes all UN members, voted 80 to 79 last month to postpone any debate on cloning for two years, virtually blocking the US-led initiative.
Encouraged by the one-vote margin of defeat, the United States and key supporters such as Spain, Portugal, Italy and Ireland, are confident they can get a majority if another vote is held.
Costa Rica's ambassador Bruno Stagno, chairman of the anti-cloning coalition, introduced the draft resolution on Friday.
It is on a crowded General Assembly agenda for late Monday, but could be delayed until Tuesday.
"US approach would ban efforts to use cloned embryonic stem cells to try and generate healthy tissues, or to treat degenerative diseases such as Parkinson's"
Executive publisher of Science
Opponents of the US position, including Britain, France, Japan, and South Africa and more than 30 nations, want a partial ban that would prohibit cloning of human beings, but allow the use of cells cloned from human embryos for research.
But supporters of a broad ban portray such "therapeutic" cloning as the taking of human lives.
The earlier motion to postpone the negotiations in the legal committee was introduced by Iran on behalf of the 57-nation Organisation of the Islamic Conference.
Islamic countries say they want more time to consider the issue. They have been more open to the practice of human cloning for medical purposes.
Some diplomats said a compromise might emerge to defer drafting of the treaty for one year rather than two.
The US anti-abortion movement is lobbying strongly for the total ban, while many scientists want the right to pursue therapeutic cloning.
Washington's approach would ban "efforts to use cloned embryonic stem cells to try and generate healthy tissues, or to treat degenerative diseases such as Parkinson's," said Alan Leshner, who heads the American Association for the Advancement of Science and is executive publisher of its journal, Science.
The US Congress is also divided over stem cell research and so far has failed to adopt legislation regulating it.