"I urge the LTTE [Liberation Tigers of Tamil Eelam], within the next 48 hours to allow free movement of civilians to ensure their safety and security. For all those civilians, I assure a safe passage to a secure environment," he said in a statement.
Government forces have pushed the Tamil Tigers into a 300sq km pocket of territory in Mullaittivu district.
But humanitarian relief organisations say that about 250,000 civilians are still trapped in the rebel-held area.
Speaking from northern Sri Lanka on Thursday, Tony Birtley, Al Jazeera's correspondent, said: "The United Nations agency in Sri Lanka is now seriously concerned for the well-being of civilians in the small and ever-shrinking pocket in the north of the country, which is controlled by the Tamil Tigers."
"The army and air force will use its aerial power and reconnaissance ability when they move into thick jungle around Mullaittivu," he said.
"At the moment, the government forces are fighting on conventional lines, as we understand. We have seen a number of army casualties.
"The army maintains it is inflicting more damage on the LTTE but we only have their word for it."
The LTTE has accused the army of shelling a no-fire zone it set up last week for civilians.
TamilNet, a pro-LTTE website, on Wednesday said 23 civilians were killed and 121 wounded, quoting unidentified medical sources.
But the Sri Lankan government insists there have been "zero civilian casualties" in its operations, and that the LTTE has moved its artillery into populated areas.
Mahinda Samarasinghe, Sri Lanka's minister for disaster management, admitted that the army could have fired into the safe zones.
"If the LTTE has positioned their long-range weapons in a corner of the safe zone ... if we know that there are no civilians in that area, then naturally we have to take on those targets. There is nothing to prevent that kind of thing happening, even under international law," he told Al Jazeera.
"I can assure you that we have not jeopardised civilians who are in the vicinity of the safe zone and we will continue to observe that principle of safeguarding civilians."
UN staff working in the war zone have witnessed the shelling of areas populated by civilians, Weiss told Al Jazeera.
"Our staff were in the area over the weekend and were in bunkers when incoming shells killed and wounded dozens of civilians," he said.
"There are a quarter of a million inside the area [which is] about a third of the size of London. They are surrounded by government troops who are fighting a fierce battle with Tamil Tiger rebels. It is really a battle to the end.
"It is most important that the warring parties exercise all caution to preserve the lives of civilians and, if possible, let those civilians go."
Jacques de Maio, the head of Red Cross operations for South Asia, said on Wednesday: "People are being caught in the crossfire, hospitals and ambulances have been hit by shelling and several aid workers have been injured while evacuating the wounded.
"It's high time to take decisive action and stop further bloodshed because time is running out."
The military says the LTTE, which claims to be fighting for the creation of an independent Tamil state in northern Sri Lanka, wants to create a "last-minute civilian tragedy" because the army was about to completely defeat them.
Udaya Nanayakkara, a military spokesman, said claims of civilian deaths are part of a "cheap propaganda exercise" by the LTTE.
The government has said civilians are being forced to move with the Tigers to act as human shields, but there has been no independent confirmation that is the case.
The Red Cross' De Maio said: "When the dust settles, we may see countless victims and a terrible humanitarian situation unless civilians are protected and international humanitarian law is respected in all circumstances."
On the war front, Sri Lankan forces pushed into the northern village of Visuamadu, held by the LTTE, on Wednesday, a defence ministry official said.
"It is high time to take decisive action and stop further bloodshed because time is running out"
Jacques de Maio, the head of Red Cross operations for South Asia
He said the Tigers withdrew their long-range artillery guns into an area declared a 35sq km "safe zone" for civilians, and were firing at the military from there.
The military seized Mullaittivu on Sunday, which they said was the LTTE's last urban centre.
A fresh offensive against the LTTE was launched by the government late last year, shrinking the northern territory under its control.
Thousands of people have been killed in Sri Lanka since the LTTE launched its war in 1972.