The United States and its partners have ruled out military action, but they do want the U.N. to endorse a plan calling on Syrian President Bashar Assad to hand over power.
Russia has threatened to veto the resolution, but the U.S. is working to get Russian leaders to back the latest version.
The move comes as Syria's current regime steps up its assault against army defectors and protestors.
More than 5,400 people have been killed since March.
The Associated Press contributed to this report.