Fabius also said that talks were under way with Russia on Syria's future if President Bashar al-Assad was ousted. But Russian Foreign Minister Sergei Lavrov swiftly denied such talks had taken place with France or the United States.
"There were no such discussions and there could not have been such discussions. This completely contradicts our position," Lavrov told reporters. "We are not involved in regime change."
Russia, along with China, has vetoed two Security Council resolutions against Assad and has vowed to oppose any military intervention.
In an interview with France Inter radio, Fabius said major powers could hold a conference at the end of the month in Geneva on the crisis which erupted in mid-March 2011 and has since cost more than 14,400 lives, according to the Syrian Observatory for Human Rights.
"There is a possibility of holding a conference in Geneva on June 30," he said. Participants would include UN Security Council countries, but the meeting would be held "without the constraints of the Security Council," he added.
Meanwhile, Syrian opposition leaders were meeting in Istanbul on Friday in a bid to
settle their differences and forge a united front to confront the escalating conflict in their homeland.
"We will work towards a unified vision," Burhan Ghalioun, the former head of the main opposition group, the Syrian National Council, told the French news agency shortly before the two-day gathering kicked off.