A military spokesman told AFP around 10,000 civilians had managed to flee to government areas, and that there was now "hardly anybody left" in the rebel zone on the northeast coast.
"They are slowly giving up. They are blowing up whatever arms and ammunition they have," Brigadier Udaya Nanayakkara said of the remnants of the once-powerful separatist army.
The navy said it had captured the family of a top Tamil Tiger military commander -- Sea Tiger chief Colonel Soosai -- as they tried to escape by boat.
But there was no sign of rebel chief Velupillai Prabhakaran.
President Mahinda Rajapakse also vowed that "all territory will be freed from Tiger control" by Sunday morning.
There was no immediate word from the rebels, but the pro-Tiger Tamilnet website said the narrow beach and lagoon area from where the rebels have been mounting a last stand was was engulfed in smoke.
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It said close-quarter combat had been raging since Friday morning.
The massive final push against