Russia has agreed to negotiate on the draft, said the official, who spoke on condition of anonymity in accordance with department rules.
The Syrian uprising, which began with mostly peaceful protests, has become increasingly violent in recent months as army defectors clash with government forces and some protesters take up arms to protect themselves.
The violence has inflamed the sectarian divide in the country, where members of Assad's Alawite sect dominate the regime despite a Sunni Muslim majority.
Assad's regime has warned that the turmoil will throw Syria into chaos, religious extremism and sectarian divisions, a message that resonates among Alawites and minority Christians who fear reprisals from the Sunni majority.
On Tuesday, Syrian reporters were taken north of Damascus to see the Sednaya Convent, believed to have been build in A.D. 547. The site was damaged by artillery fire Sunday, in an attack the government blamed on "armed terrorists." No casualties were reported.
"Providence has salvaged this holy site," said Sister Verona, the head of the Sednaya Convent.
Also Tuesday, army defectors gained full control of the central town of Rastan after days of intense clashes, according to a town activist who identified himself as Hassan. He refused to give his full name, fearing reprisal.
The town was taken by defectors twice in the past only to be retaken by Syrian troops. Rastan is the hometown of former Defense Minister Mustapha Tlass, who held the post for more than three decades, mostly under Assad's father and predecessor, the late Hafez Assad.
AP writers Anita Snow at the United Nations, Nataliya Vasilyeva in Moscow and Dale Gavlak in Amman, Jordan, contributed to this report.
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