The Council in July agreed in a resolution to allow truckloads of much-needed aid to cross into rebel-held Syrian territory without the consent of the Damascus regime.
Australian Ambassador Gary Quinlan, whose country chairs the 15-member council this month, said his country along with Luxembourg and Jordan will move quickly to seek a 12-month extension of the aid deliveries.
"We will be having consultations urgently over the next couple of weeks about extending that mandate for 12 months," said Quinlan.
The UN's top humanitarian aid official Valerie Amos asked the council to renew authorization for the aid deliveries that are due to end in January.
While the convoys have not reached as many people in need as hoped, "they have made a difference," said Amos, the Under-Secretary General for Humanitarian Aid.
Over the past six months, 30 UN convoys carrying food aid, medical supplies, water and sanitation equipment have been sent from Turkey and Jordan.
The opposition Syrian National Coalition said in a statement that the convoys had "reached only a fraction of the 3.5 million it was intended to assist" and called on the UN to ramp up deliveries.
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The Australian ambassador told reporters following closed-door council consultations that "there is a sense that we will be able to scale up."
The nearly four-year war in Syria has forced almost half of Syrians to flee their homes, many of them multiple times.
There are now 7.6 million people displaced inside Syria and 3.2 million others have fled the country, mostly to bordering nations.
"This is the largest number of people displaced from conflict in the world," Amos said.
Some 12.2 million Syrians are in need of aid, up from 10.8 in July.
Amos asked the council to press all sides in the Syrian war to stop blocking aid deliveries and to push for an end to the violence.
A UN report this month said that 10 requests by the World Health Organization to deliver aid since October had gone unanswered by the government of Bashar al-Assad.
UN envoy Staffan de Mistura has proposed setting up special zones in Syria to "freeze" fighting to allow for humanitarian aid deliveries and to create space for political negotiations.