The draft, proposed by the UK, France, Germany and Portugal and backed by the US, threatens sanctions only if the repression of protests does not end.
The scaled-back version is aimed at winning the support of China and Russia, which oppose sanctions.
Western diplomats say they are planning for a vote by the end of the week.
The draft resolution “demands an immediate end to all violence”, according to copies obtained by news agencies.
It says the 15-nation Security Council “expresses its determination, in the event that Syria has not complied with this resolution, to adopt targeted measures, including sanctions”.
Last month the US, Britain, France, Germany and Portugal put forward a draft resolution calling for sanctions against President Assad, members of his family and close associates.
But Russia and China have threatened to veto any sanctions resolution against Syria.
Others countries which oppose sanctions include a bloc of key emerging powers – India, South Africa, and Brazil.
The BBC’s Barbara Plett at the UN in New York says some countries fear any outside involvement in Syria could trigger a much wider crisis.
Their position has been hardened by Nato’s actions in Libya, which they say has overstepped a UN mandate to protect civilians there, she adds.
One European diplomat told Reuters news agency that they wanted to send “a strong and unified message to ensure that the Assad regime does not remain deaf to the demands of the international community”.
Meanwhile, Syrian forces are continuing efforts to suppress protests that began six months ago.
To read more, visit:Â http://www.bbc.co.uk/news/world-middle-east-15085816
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