There was no response to the deadline from the LTTE.
Lakshiman Hulugalle, a Sri Lankan defence ministry spokesman, told Al Jazeera that a post-deadline military plan had not yet been finalised.
"What we have requested the LTTE leaders and LTTE cadres is to lay down arms and surrender themselves. So when we start operations, the LTTE leaders have to face that and they have no option but getting killed," he said.
International organisations continue to highlight the vulnerability of citizens in the area, whose safe passage out of the region the government had promised on Monday.
Human Rights Watch (HRW) said on Tuesday that between 50,000 and 100,000 people remain stranded in a 17km square "no-fire zone" - the last region controlled by the LTTE.
Brad Adams, HRW's Asia director, said: "Both sides need to show far greater concern for civilians, or many more civilians will die."
Sarah Crowe, a Unicef spokesperson, told Al Jazeera: "Things are on a knife edge at the moment. It is a human avalanche.
"Some 65,000 people are crammed into overcrowded camps. What we're concerned about now is, with this new flood of people coming in, the camps will be so overcrowded that they will be unable to cope.
"We know that water and sanitation are dire and that children are already quite malnourished."
However, Hulugalle said that the government could cater for the refugees.
"We have been looking after more than 70,000 refugees for the last couple of months,' he said.
"Their welfare is being looked after. Once their registration is done their basic needs are being looked into."
The government had warned Velupillai Prabhakaran, the LTTE leader, that he had to give up by the deadline or face a "military course of action".
The threat came after the government said its troops had advanced into the last part of Tiger-held territory after breaching an earth barricade.
Lucien Rajakarunanayake, a presidential office spokesman, told Al Jazeera on Monday: "The government advanced to a 3km earth barrier on the edge of the no-fire zone and destroyed it this morning.
"I suppose the government will be in a position to declare military victory very soon."
The Sri Lanka military said suicide bombers had blown themselves up among civilians as they were leaving the area, but as no journalists are allowed into the area his claim could not be verified.
Nanayakkara said: "At least 17 civilians, including women and children, have been killed and 200 people injured from the cowardly suicide attacks."
Al Jazeera's David Chater, reporting from Colombo, the Sri Lankan capital, said defence ministry officials had told him there were about 300 Tamil Tigers armed with heavy weapons left in the zone.
"As the civilians flee, [the military says that] the suicide bombers are inside the groups and detonate their devices," Chater reported.
There was no immediate comment from the LTTE about the reports.
David Poopalapillai, a spokesman for the Canadian-Tamil Congress which works for advocacy for Tamils, told Al Jazeera from Canada: "It is very bleak. The Sri Lankan government, as we speak, are committing war crimes on Tamil communities in Sri Lanka.
"We have a credible report today from the war zone that 14,000 civilians had been massacred," Poopalapillai said.
"Please understand that this war is being conducted without any witnesses, without any media. If Sri Lanka is genuine, if they are really telling the truth, why don't they allow agencies and other international media to go to the war zone and report from that spot?
"The civilians are very scared to go to the government side. They feel that when they go to the government side they will be put into internment camps ... that is the reason why they are staying in this safe zone."
The UN has repeatedly accused the military of shelling the "no-fire" zone, and the Tamil Tigers of preventing civilians from leaving, in effect using them as human shields.
Both sides reject the claims.
The UN says at least 4,500 civilians have been killed in the past three months of heavy fighting and another 12,000 injured.
A 2002 truce all but collapsed in 2005, and the military said since late 2007 that it aimed to "wipe out" the LTTE in 2008.
The LTTE, commonly known as the Tamil Tigers, has been fighting for a Tamil homeland in the north of Sri Lanka, arguing that ethnic Tamils are marginalised under the Sinhalese majority government.