The Red Cross says it has been refused permission to deliver aid to the Baba Amr district of the bombed-out Syrian city of Homs, despite earlier getting the go-ahead from the authorities.
ICRC President Jakob Kellenberger said the hold-up was "unacceptable".
Baba Amr has suffered heavy bombardment by government forces in recent weeks.
The UN secretary general said the Syrian authorities had committed clear and widespread crimes against the country's civilian population.
In an address to the General Assembly, Ban Ki-moon said the international community had failed in its duty, and inaction had encouraged Syria's leaders in their repression of civilians.
"The images which we have seen in Syria are atrocious," he said. "It's totally unacceptable, intolerable. How, as a human being, can you bear this situation?"
The Syrian ambassador to the UN, Bashar Jaafari, said opposition fighters had been using civilians as human shields in Homs.
The rebel Free Syrian Army (FSA) said on Thursday it was leaving the district in a "tactical withdrawal".
On Friday the UN human rights office said it had received reports of a "particularly grisly set of summary executions" of 17 people in Homs.
Meanwhile Paul Conroy, a Sunday Times photographer who fled Syria after being wounded in Homs, told the BBC that what was happening in Baba Amr was "systematic slaughter".
Two French journalists caught up in the shelling and smuggled out of Homs into Lebanon have been flown back to a military airport outside Paris.
Edith Bouvier and William Daniels were met on arrival by President Nicolas Sarkozy.
Ms Bouvier was badly injured in the bombardment of a makeshift media centre last week, in which two other journalists were killed.
She was stretchered off the plane and is set to undergo surgery on Friday evening for multiple leg fractures.
The bodies of the two dead journalists, Marie Colvin of Britain's Sunday Times and French photojournalist Remi Ochlik, have arrived in Damascus and are expected to be returned home.
Mr Kellenberger said in a statement that the seven-lorry aid convoy carrying food, medicine and blankets, along with ambulances from the Syrian Red Crescent, would stay in Homs overnight in the hope of entering Baba Amr "in the very near future".
"It is unacceptable that people who have been in need of emergency assistance for weeks have still not received any help," he said.
The statement added that the Syrian authorities had earlier given a "green light" for the convoy to enter, and that the problem was not a technical hitch but something more serious.
The convoy had also been hoping to evacuate the wounded.
Mr Kellenberger said that in the meantime the group would help those families that had fled Baba Amr.
But the BBC's Jim Muir in neighbouring Lebanon says the tone of the statement suggests that the Red Cross is not very confident about getting permission to enter Baba Amr immediately on Saturday either.
Of the 100,000 people who normally live there only a few thousand remain, with the FSA saying it had pulled back to save those still there from an all-out assault.
Many of those still in the district are without power and running low on basic supplies. The ICRC has said it fears there could be many seriously wounded people there.
The delay has given rise to opposition allegations that government forces were removing evidence of summary killings.
The opposition Local Co-ordination Committees reported that in Syria as a whole 56 people had died on Friday, of which 32 were killed in Homs and 16 in the nearby town of Rastan.
Activists spoke of revenge killings in an agricultural area outside Homs, and the summary killing of 10 people behind a local co-operative building.
The reports prompted UN High Commissioner for Human Rights spokesman Rupert Colville to warn the Syrian government of its responsibilities under international law.
"Enough crimes have already been committed in Syria over the past year," he said.
"We urge the authorities to make sure no more are committed now that they have taken control of Baba Amr."
The UN estimates more than 7,500 people have died in the 11-month anti-government uprising in Syria.
'No place to hide'
Mr Conroy, who was smuggled out of Syria into Lebanon on Tuesday, described the scenes in Homs from his hospital bed in the UK.
"I've done a fair few wars, I've never seen anything on this level," he said.
"There are no targets, it's pure systematic slaughter of a civilian population."
Spaniard Javier Espinosa, who also escaped, described his flight as part of a group of 50 people who crept through the government lines at night.
"There were a group of kids who were terrified... we tried to just shut [quieten] them down ... but it was too late and they [government troops] started shooting, so we had to run for our life... to hide," he told the BBC.
"I guess there were some people who died."
He also spoke about the suffering he saw while he was in the city.
"We are talking about 20,000 mainly women and old people, civilians, trapped in a very small enclave under constant shelling during the whole day until night," he said.
"It was very systematic. They start shelling at six o'clock in the morning, they finish at six o'clock pm. So there is no place to hide because there are no shelters, just wait in your house [and hope] that they don't hit your house. And there is no basic stuff like milk for the babies, like bread, like water."