The government said thousands had fled the area in the north
More than 10,000 Sri Lankan civilians have fled in the past few days from rebel-held territory in the north, the government says.
Sri Lankan President Mahinda Rajapakse called on the Tamil Tigers to let thousands more civilians leave and then unconditionally surrender.
The rebel-held enclave is coming under sustained pressure from the army.
Aid agencies have expressed grave concern for the more than 200,000 civilians believed trapped in the area.
The government has accused the rebels of preventing the non-combatants from leaving and using them as human shields.
But the Tigers say the civilians are staying in the enclave because they fear the army.
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 Rajapakse becomes president
2006: Heavy fighting resumes
2009: Army takes main rebel bases of Kilinochchi and Mullaitivu
Independent journalists are prevented by the government from travelling to war-hit areas, so there is no way of getting independent corroboration of either side's claims.
The defence ministry said in a statement carried by AFP news agency: "Over 10,000 civilians have come to Kilinochchi while 139 others have come to Jaffna since [Wednesday].
"Among the rescued civilians are over 2,800 children and about 3,000 women."
Meanwhile, the army has said it was closing in on remaining Tamil Tiger positions, confining them to an area of about 200 sq km (124 sq miles).
With fighting intensifying, the military said the Tigers suffered "heavy damages" on Saturday.
The pro-rebel Tamilnet website said 120 civilians had been killed in two days of shelling.
Aid agencies - most recently the UN's World Food Programme (WFP) - continue to remain concerned about the plight of thousands of civilians who it is feared are trapped between the two sides.