Mahinda Rajapaksa, Sri Lanka's president, on Thursday promised safe passage to all civilians and gave the Tamil Tigers until Saturday night to let them leave.
But Sophie Romanens, a spokeswoman for the International Committee of the Red Cross, told Al Jazeera that few people appeared to have been able to flee as clashes between the sides continued.
"Our team on the ground could still here fighting there and didn't notice any significant movement of population out of the conflict area," she said from Colombo, the capital.
"We are very concerned about the situation of civilians ... it is very important to understand that there has been massive displacement in the last few months."
The Red Cross says hundreds of civilians, including children, have been killed or wounded in fighting since last week. Colombo disputes the figures.
After the apparent failure of the ceasefire, Sri Lankan military officials said that it would have to move into the area to help them.
"We will now have to save the civilians," Kaheliya Rambukwella, the defence ministry spokesman, said on Sunday.
"It is now very evident that Prabhakaran [the Tamil Tiger leader] is ... using civilians as cover. We will take the utmost care of civilians when we move in," he said.
The government says the Tamil Tigers are preventing the civilians from leaving, but the LTTE has said people are refusing to go as they fear being abused by the army.
S. Pasupathi, the co-ordinator of the World Tamil Relief Fund, told Al Jazeera that Tamil civilians "simply don't trust the Sri Lankan government".
"It is impossible for the LTTE with a small number of soldiers to hold onto 250,000 Tamils," he said.
"I think they feel safer in LTTE-controlled areas than the safe zone or army-controlled areas."
The defence ministry said that the separatists were fighting around the government-declared 35sq km "safe zone" where civilians had been encouraged to take shelter.
"Terrorists are manning a forward defence line in the outer perimeters of the safe zone, which is also said to be cluttered with land mines clamping any civilian attempt to reach the liberated areas," it said.
The LTTE has accused the military of shelling the area in and around what the government says is a safe zone.
Meanwhile, the Sri Lankan military said it had captured two camps used by Tamil Tiger suicide squads and seized a large number of weapons.
"It is a big success because that means the Tigers are leaving their camps and fleeing," Brigadier Udaya Nanayakkara, a military spokesman, told the Associated Press news agency.
The claims cannot be independently verified as the war zone is closed to journalists and there was no immediate comment from the LTTE.
During months of heavy fighting, government troops have vastly reduced the area under Tamil Tiger control, capturing their de facto capital of Kilinochchi and the Tamil Tiger bastion of Mullaittivu on the northeast coast.
More than 70,000 people have been killed in the LTTE's struggle for an independent homeland in the north and east of the island.
Ethnic Tamils have long complained of marginalisation at the hands of successive governments controlled by the Sinhalese majority.