The resolution would seek to re-launch a political dialogue that broke down after Yemen's Shiite Huthi rebels pressed ahead with an offensive, forcing President Abedrabbo Mansour Hadi to flee to Saudi Arabia.
The six-nation Gulf Cooperation Council (GCC) has been negotiating with the five permanent Security Council members and Jordan on a text after Saudi Arabia launched an air campaign on Yemen on March 23, without a UN mandate.
The GCC is no longer seeking a resolution that supports Saudi-led military action in Yemen, which it argues is legal because it is being carried out at Hadi's request, diplomats said.
Saudi Arabia accuses Iran of backing the Huthis in their power grab as part of a bid to extend its regional influence.
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"The GCC will now need to lobby the Russians very, very hard," said a Security Council diplomat, who spoke on condition of anonymity.
During negotiations, Russia presented amendments to the draft resolution that would extend an arms embargo to all sides, including Hadi's forces in the conflict, diplomats said.
Moscow also opposed sweeping sanctions against the Huthis and requested that a list be submitted of individual names of rebel leaders who could be targeted for a global travel ban and assets freeze.
Talks on how to address the crisis in Yemen came amid mounting UN concern over the civilian toll from the seven-day Saudi-led air campaign.