Fabius told reporters at UN headquarters in New York that discussions on a text would start "in the coming days."
The Security Council in December rejected a resolution that would have set a deadline for reaching a final peace deal and pave the way to the creation of a Palestinian state.
The United States had voted against the measure but was spared from resorting to its veto after eight council members including France voted yes, one vote short of the nine needed for adoption.
"I hope the partners that were reluctant will be less reluctant," Fabius said.
"It is necessary to move forward to have a solution to this problem," he added.
- US cautious -
The United States offered a cautious reaction to the French plan.
"We're not going to get ahead of any decisions about what the United States would do with regard to potential action at the UN Security Council," a US official told AFP.
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"We continue to engage with key stakeholders, including the French, to find a way forward that advances the interest we and others share in a two-state solution," said the official, who spoke on condition of anonymity.
The French move came a day after UN envoy Robert Serry told the Security Council that it should step in to present a "framework for negotiations, including parameters" to achieve peace.
"This may be the only way to preserve the goal of a two-state solution, in the present circumstances," he said in a bluntly worded assessment of the Israeli-Palestinian crisis.
International concern over the fate of the peace process spiked after Israeli Prime Minister Benjamin Netanyahu vowed during his election campaign that he would never allow the establishment of a Palestinian state under his watch.
Netanyahu later backtracked on his comments but the US administration appeared unconvinced and UN Secretary General Ban Ki-moon urged him to renew his commitment to a two-state solution.
Fabius stressed that a UN draft resolution could be presented to the 15-member council "in the coming weeks."
Serry warned that a new UN resolution to re-launch negotiations would be fruitless without a genuine commitment from both sides to reach a deal.
"If the parties are not ready to negotiate it would be wrong to rush them," he said.
But Fabius stressed that UN action could help nudge the two sides back to the negotiating table.
"Obviously the two parties must discuss, but the discussion must be backed by an international effort," he said.
Israel has long maintained that direct talks with the Palestinians are the best framework for advancing peace talks and has bristled at UN involvement to set a timeframe for a deal.