If the text proposed by France is accepted, the MINUSMA mission in Mali would provide medical evacuations, fuel, water, rations, and engineering units to assist the G5 Sahel troops provided by Burkina Faso, Chad, Mali, Mauritania and Niger.
The G5 conducted its first operation last month in the Sahel region, which has turned into a hotbed of lawlessness since chaos engulfed Libya in 2011, Islamists took over northern Mali in 2012, and the Boko Haram group became active in northern Nigeria.
French President Emmanuel Macron has complained about the sluggish deployment of the G5 Sahel and said it needed more troops and military means.
He has called an international meeting in Paris next Wednesday in an effort to address the shortfalls.
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In its proposed resolution, France asked United Nations Secretary General Antonio Guterres to conclude "as soon as possible" a technical agreement between the UN, the European Union and the G5 Sahel states "with a view to providing specified operational and logistical support through MINUSMA" to the new force.
MINUSMA has almost 11,000 Blue Helmets and has been deployed in Mali since 2013.
France has its own 4,000-strong military presence in the region, known as Barkhane.
US Ambassador to the UN Nikki Haley has said that the peacekeeping mission in Mali was already overstretched. She raised "serious" reservations about "using UN resources to support non-UN activity."
Financing of the G5 Sahel force, which Paris says would need 400 million euros (USD 473 million) when it is fully operational next year, is not yet assured.