A text needs support from nine nations on the 15-member U.N. Security Council to be voted upon. The French official spoke on condition of anonymity.
The Arab League wants the world body to pass a resolution condemning Syria for systematic violence against the country's population. The U.N. estimates about 5,400 people have been killed in 10 months of violence.
Meanwhile, Russia's Foreign Ministry on Monday said it has invited Syrian authorities and opposition for talks in Moscow.
In a statement released Monday, the ministry said that Syrian authorities have already agreed to come. The ministry is hoping that opposition leaders will send their reply in the coming days. The opposition has balked at holding talks with the regime, saying the violence must end first.
The ministry said the Syria talks need to be conducted "as soon as possible" to stop violence in the country.
Russia -- Syria's longtime ally -- has been backing the regime of President Bashar Assad although Moscow has also talked to Syrian opposition leaders in the past months.
In London, British Prime Minister David Cameron's office urged Moscow to reconsider its opposition to the measure.
"Russia can no longer explain blocking the U.N. and providing cover for the regime's brutal repression," a spokeswoman for Cameron said, on customary condition of anonymity in line with policy.
Russia insists it won't support any resolution it believes could open the door to an eventual foreign military intervention in Syria, the way an Arab-backed U.N. resolution paved the way for NATO airstrikes in Libya. Instead, the Kremlin said Monday it was trying to put together negotiations in Moscow between Damascus and the opposition.
In new violence on Monday, activists said Syrian forces heavily shelled the restive city of Homs and troops pushed back dissident troops from some suburbs on the outskirts of Damascus in an offensive trying to regain control of the capital's eastern doorstep.
Assad's regime is intensifying its assault aimed at crushing army defectors and protesters, even as the West tries to overcome Russian opposition and win a new U.N. resolution demanding a halt to Syria's crackdown on the 10-month-old uprising. Activists reported at least 28 civilians killed on Monday.
Also on Monday, Secretary of State Hillary Rodham Clinton announced that she'll attend a U.N. Security Council meeting Tuesday to lend her support to Syrians seeking democratic change.
Clinton says the U.S. will support an Arab-backed condemnation of President Assad's regime. She says Assad is blocking a peaceful transition by attacking the Syrian people, and warned that instability could spill over throughout the Mideast.
Clinton's statement condemned the Assad regime's escalation of violence "in the strongest possible terms." She called the shelling of civilian areas with mortar and tank fire "brutal" and said whole buildings were leveled.
The Associated Press contributed to this report.