"I don't want him to take cyanide and commit suicide. He has to face charges for his actions."
The ultimatum comes after months of heavy fighting. 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.
Last Tuesday, Lieutenant Sarath Fonseka, a Sri Lankan army chief, also vowed to end the Tigers' aim of a separate state.
"Our objective should be to eliminate the terrorism and the LTTE's 'Eelam Dream' entirely from the country," Fonseka said.
Rajapaksa had said on state television on Monday that government forces had opened routes to enable about 35,000 people trapped by the fighting to flee in what he said was the "largest-ever hostage rescue mission".
Lucien Rajakarunanayake, a presidential office spokesman, told Al Jazeera: "The government advanced to a 3km earth barrier on the edge of the no-fire zone and destroyed it this morning.
"At least 13,000 Tamil civilians who were held as hostage by the LTTE in the no-fire zone have escaped into the government-held safety zone.
"We are told there are more people coming ... I suppose the government will be in a position to declare military victory very soon."
Suicide bombers had blown themselves up among civilians as they were leaving the area, a military spokesman said.
Brigadier Udaya 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 "no-fire 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. It is not possible to independently confirm the military's reports as journalists are banned from the area.
David Poopalapillai, a spokesman for the Canadian-Tamil Congress, 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."
Aid groups believe more than 100,000 civilians have been trapped between the army and Tamil Tiger fighters in the "no-fire zone".
The UN has repeatedly accused the military of shelling the area, 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 fighting and another 12,000 injured.