French President Emmanuel Macron will attend a summit in Bamako in July to consider how the Sahel force will work .
A draft resolution circulated to the 15-member council would give the Sahel force a UN mandate to "use all necessary means" to "combat terrorism, drug trafficking and trafficking in persons," according to the text obtained by AFP.
The Security Council could vote on the draft resolution as early as next week, diplomats said.
Burkina Faso, Chad, Mali, Mauritania and Niger -- which make up the G5 -- agreed in March to set up the special counter-terrorism operation of 5,000 troops for the Sahel region.
The force will have its headquarters in Mali, but will be under a separate command from the UN peacekeeping force MINUSMA, which has been deployed in the country since 2013.
French Ambassador Francois Delattre said he expected the council to back the measure, drafted under chapter 7 of the UN charter, which provides for the use of force.
"We cannot afford to let the Sahel region become a new safe haven for terrorists from across the region," he told reporters.
The draft resolution would authorize the deployment of the force of up to 5,000 military, police and civilian personnel for an initial period of 12 months.
The force will operate in border areas and work to "create a secure environment by eradicating the actions of the terrorist armed groups and other organized criminal groups," the draft said.
France carried out a military intervention in Mali in 2013 to drive out jihadist groups, some of which were linked to Al-Qaeda, which had seized key cities in the country's north.
Although the Islamists have been largely ousted from the north, jihadist groups continue to mount attacks on civilians and UN forces in violence that has engulfed parts of central Mali.
The proposed measure includes a request to UN Secretary General Antonio Guterres to report to the council within 60 days on ways to support the Sahel force through logistical and financial means.
UN support for the Sahel force would come at a time when the 12,000-strong MINUSMA peacekeeping force has come under a series of attacks in what has become the UN's deadliest mission.
Two peacekeepers from Chad were killed in an ambush in the north on May 23.
The attack, like many others in the last weeks, was claimed by a powerful jihadist alliance, the Group to Support Islam and Muslims (GSIM), a fusion of three Malian jihadist groups with previous Al-Qaeda links.
The French-drafted text stressed that the G5 force's efforts to counter jihadist groups will help MINUSMA fulfil its mandate to stabilize Mali.
Speaking at a meeting in the Malian capital Bamako, Niger's Foreign Minister Ibrahim Yacouba said the G5 hoped to have a UN mandate "by the end of June," paving the way to deployment of the force.
French President Emmanuel Macron will attend a summit in Bamako in July to consider how the Sahel force will work alongside France's own 4,000-strong military presence in the region.
"France is an important partner," said Chad's Foreign Minister Hissein Brahim Taha.