The world body on Monday may urge each member government to adopt its own laws on human cloning.
At the heart of the debate are stem cell studies and other research that rely on therapeutic cloning, in which human embryos are cloned to obtain the cells used in the studies and are later discarded.
Many governments, particularly those with large Catholic populations, say they view this type of research, for whatever purpose, as the taking of human life.
The UN project dates back to 2001, when France and Germany proposed a worldwide ban on human cloning by way of a binding global treaty.
That attempt failed after Washington fought to broaden the ban to all cloning of human embryos, a step many scientists and governments argued would block some promising avenues of medical research.
The US campaign to persuade the 191-nation UN General Assembly to approve a broad anti-cloning treaty ran out of steam last November when the assembly's treaty-writing legal committee, after years of debate, remained deeply divided on
Opponents of the US plan said the outcome showed that a majority of UN members, while they supported a ban on the cloning of human beings, wanted to keep the door open to scientific studies relying on therapeutic cloning.
As a face-saving compromise, Italy suggested that a committee working group reconvene in February for talks on a non-binding political declaration in place of a treaty.
Rome suggested the declaration urge nations to adopt laws "to prohibit any attempts to create human life through cloning processes and any research intended to achieve that aim".
While proponents of embryonic stem-cell studies objected to use of the term human life, all sides agreed the text could form the basis of future negotiations.
The working group's goal is to agree on a draft to be submitted to the legal committee by the end of this week.
The working group consists of all 191 assembly members, so whatever it decides on presumably would be acceptable to the full assembly as well.
But diplomats warned the week would be a difficult one. "You're trying to arrive at a consensus on a matter on which there is no consensus," said one diplomat.
"The issue is very polarised and it's quite a big gap to bridge. But one way or another, it will be resolved by the end of the week - we hope by consensus but if not, through a vote," the diplomat said.
Moroccan ambassador to the UN, Muhammad Binnuna, who will be chairing the closed-door meeting, said he had been quietly working since November to bring the sides together.
Though no agreement had yet been reached, he said: "I think it is possible."