After the council unanimously passed the resolution on Syria's chemical arsenal on Friday, Australian UN Ambassador Gary Quinlan appealed for members to quickly take action on the "ever-accelerating humanitarian catastrophe" in Syria.
"Each day we delay creates another 6,000 refugees," said Quinlan, who is president of the Security Council for September. He called for the council to try to reach agreement on a statement this week.
More than 2 million Syrians, mostly women and children, have fled during the 2-1/2-year-old foreign-sponsored insurgency, which the United Nations says has killed more than 100,000 people.
Millions more inside Syria are in desperate need of help but aid has slowed to a trickle due to excessive red tape and violence.
12 international aid groups are approved by the Syrian government to work in the country and convoys of aid trucks struggle to meet demand, delayed by having to negotiate dozens of checkpoints, many of them imposed by militants.
Several aid convoys have also been targeted by militants.
Deputy UN council envoys are due to meet to discuss the proposed Security Council presidential statement on Monday, said diplomats, speaking on the condition of anonymity. Unlike a resolution, a presidential statement is not legally binding.
The draft aid text is based on a wish list that UN aid chief Valerie Amos sent the council last month that included allowing cross-border delivery, humanitarian pauses in fighting and advance notice of military offensives. Diplomats described the list as ambitious.
The draft statement urges the Syrian authorities to promptly facilitate "safe and unhindered humanitarian access to people in need through the most effective ways, including across conflict lines and, where appropriate, across borders from neighboring countries in accordance with the UN guiding principles of humanitarian emergency assistance."
However implementation of the proposal may face difficulties because it needs cross-border access which the government is worried might be used by foreign supporters of the anti-Syria militants to smuggle more arms into the country.
Syria has been gripped by a deadly unrest which is funded by several regional and Western countries, most of all Qatar and Saudi Arabia.
SHI/SHI