The draft, proposed by the UK, France, Germany and Portugal and backed by the US, removes all sanctions that has been stalled in the Security Council for months.
The scaled-back version is aimed at winning the support of China and Russia, which oppose sanctions.
The new draft does, however, threaten sanctions if Syria does not comply within 15 days with demands to cease violence against civilians and allow humanitarian agencies to enter the country.
Western diplomats say they are planning for a vote by the end of the week.
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.
Other members opposing sanctions include a bloc of key emerging powers - India, South Africa, and Brazil.
