A draft resolution presented to the 15-member council called on UN member states to "redouble and coordinate their efforts to prevent and suppress terrorist acts committed specifically by ISIL" and groups linked to al Qaeda.
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"The exceptional and unprecedented threat posed by this group to the entire international community requires a strong, united and unambiguous response from the Security Council," French Ambassador Francois Delattre said.
"This is the goal of our draft resolution, which calls on all member states to take all necessary measures to fight Daesh (IS)."
Delattre said he was seeking rapid approval of the draft resolution that was "put in blue" -- a UN term designating that a final version is ready for a vote at the Security Council.
France's bid for UN backing came after Russia submitted a revised text of a separate draft resolution that calls for fighting the IS group with Syria's consent.
That draft has been rejected by the United States, Britain and France, which are refusing to cooperate with President Bashar al-Assad's regime, whom they accuse of fomenting extremism by resorting to brutality.
The French draft resolution does not provide any legal basis for military action and does not invoke chapter seven of the UN charter that authorises the use of force.
French diplomats maintain though that it will provide important international political support to the anti-IS campaign that has been ramped up since the attacks in Paris on Friday that left 129 dead.
The French draft text describes IS as a "global and unprecedented threat to international peace and security" and said sanctions would soon be imposed against IS leaders and supporters.
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The text "unequivocally condemns in the strongest terms the horrifying terrorist attacks" by the IS group in Paris and Beirut, and also mentions violence in Tunisia, Turkey and Egypt this year.
French President Francois Hollande is due to meet Russian President Vladimir Putin in Moscow next week, two days after talks in Washington with President Barack Obama aimed at stepping up the campaign against the extremists.
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