The pro-rebel website TamilNet has accused the Sri Lankan army of firing into the government-designated "safe zone" for civilians, with more than 100 feared dead.
1976: Liberation Tigers of Tamil Eelam form in the north-east
1987: India deploys peace-keepers to Tamil areas but they leave in 1990
1993: President Premadasa killed by Tiger bomb
2001: Attack on airport destroys half Sri Lankan Airlines fleet
2002: Government and rebels agree ceasefire
2005: Mahinda Rajapaksa becomes president
2006: Heavy fighting resumes
2009: Army takes main rebel bases of Kilinochchi and Mullaitivu
There is no way of confirming reports from the conflict zone as independent journalists are barred.
Military spokesman Brig Udaya Nanayakkara told the BBC the army had no need to fire at civilian areas.
He strongly denied allegations that the military was responsible for the attacks.
There has been no word from the Tamil Tigers.
The government has also accused the rebels of using civilians as human shields.
Another UN spokesman, the resident coordinator Neil Buhne, told the Associated Press news agency that the situation was desperate.
"There have been many civilians killed over the last two days. It's really a crisis now."
He said both sides had been trying to reduce civilian casualties but added: "In the current phase, with such a level of fighting and with so many people around, unless there is very, very close attention to it, it's almost inevitable."
Mr Buhne said there had "definitely been fighting" in the safe zone.
Brig Nanayakkara said the rebels were firing from the safe zone "but we don't engage them".
There are thought to be about 250,000 civilians in the area in which the rebels are still operating.
After taking Mullaitivu, army commander Lt Gen Sarath Fonseka said the war was "95% over".
The defence ministry website's latest report says helicopter gunships had "successfully" targeted rebel positions north of Mullaitivu and that "troops of 57 Division are on the final phase of the offensive".
About 50,000 troops are involved in the offensive and navy vessels have been deployed to try to prevent escape by sea.
The government has vowed to crush the rebels, who have been fighting for a separate homeland for Tamils for 25 years. At least 70,000 people have been killed during the insurgency.
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