The draft Security Council resolution, focused on ships leaving from Libya, is currently being circulated among the five permanent members of the Security Council and the other European countries affected.
But it has not yet been distributed to the full 15-member Security Council, diplomats said yesterday.
Russian Ambassador to the UN Vitaly Churkin, this month's president of the council, said the resolution "may well be adopted in September."
Another Security Council diplomat expressed hope the resolution could be adopted "in the next couple of weeks, before the UN General Assembly" meets in late September.
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That draft died because it would have required the approval of Libya, a prospect complicated by the political tumult in the country where the internationally recognised Libyan government does not control the coastal territory.
The new resolution would authorize European navies to board and inspect suspicious vessels. If migrants are found on board, they would be given first aid and sent on to Italy, where they could try to seek asylum.
According to one diplomat yesterday, EU member states currently have differing degrees of legal leeway to act in the Mediterranean, with Italy, for instance, able operate more freely in the region than Britain or Germany.
Secretary-General Ban Ki-moon has organised a high-level meeting on migration on the sidelines of the General Assembly on September 30, to which he invited European leaders Tuesday.