In a statement released Tuesday, Fabius also called for "all the international partners who so desire to become involved in the negotiations."
The statement was issued as the Palestinian Authority is preparing to present, through Jordan, a Security Council resolution setting a two-year deadline to return the territory taken by Israel in 1967. The resolution could be put to a vote as early as Wednesday.
Fabius met Tuesday with Arab League Secretary-General Nabil Al-Arabi, Palestinian Foreign Minister Riyad Al-Malki and Mauritanian Foreign Minister Ahmed Ould Teguedi.
He also met separately with former Israeli President Shimon Peres after, on Monday, meeting with his British and German counterparts. Fabius also had an "airport meeting" Monday night with U.S. Secretary of State John Kerry to discuss issues in the Middle East.
As both French Assembly and Senate adopted recently resolutions calling their government to recognize Palestine' statehood, Fabius has said that France would do so if peace efforts failed.
"If this last attempt at negotiation does not succeed, France will assume its responsibilities by recognizing the State of Palestine. France recognizes the State of Palestine; it is not a favor, it is a right," he said.
Peace talks between Israeli and Palestinian negotiators broke down in April over Israel's refusal to release a group of Palestinian prisoners despite earlier pledges to do so.