"Troops have entered Visuamadu. There's heavy fighting in the area," the official said.
Government forces were now too close to the Tiger positions for the rebels to use their long-range artillery, the official said.
The territory under control of the Tigers has been rapidly shrinking since mid-2007 and the guerrillas lost their last urban centre when the military seized the town of Mullaittivu on Sunday.
During clashes on Tuesday, security forces engaged the rebels in five different areas in the small triangle of coastal territory still held by the LTTE and killed 19 rebels, the military official said.
Two of the rebels killed by troops on Tuesday were believed to be senior field commanders, the officer said. He did not say if the military suffered casualties in the latest fighting.
Government forces are trying to capture 30 kilometres (18 miles) of sea front north of Mullaittivu, the rebels' main military headquarters.Seizing the coastline would leave the remaining LTTE forces -- and possibly Tamil Tiger leader Velupillai Prabhakaran -- totally encircled by ground troops.
The continued heavy fighting came as the International Committee of the Red Cross (ICRC) sounded the alarm over what it said was a major humanitarian crisis in northern Sri Lanka, where some 250,000 ethnic Tamil civilians have been caught up in the war.
Medical facilities there were overwhelmed by "hundreds of dead and scores of wounded," the ICRC said in a statement from its headquarters in Geneva.
"It's high time to take decisive action and stop further bloodshed because time is running out," said Jacques de Maio, ICRC head of operations for South Asia.
According to a local health official, at least 178 civilians have been killed and another 725 wounded in fighting during January.
The pro-rebel Tamilnet.com website says the civilian toll is far higher.
Sri Lanka has dismissed charges of widespread civilian deaths with military spokesman Brigadier Udaya Nanayakkara branding such claims part of a "cheap propaganda exercise" by the Tigers.
Meanwhile, the state-run Daily News said Tamil Tigers prevented international aid workers from evacuating 300 hospital patients inside rebel-held territory.
A convoy of two dozen vehicles arranged by the ICRC and the United Nations to transport the sick was barred from crossing the frontline, the daily said.
The patients from the Puthukkudiriruppu area in Mullaittivu district were to be handed over to the military and transferred to a hospital in Vavuniya, which lies in a government-controlled area in the north.
The Red Cross spokeswoman in Colombo, Sarasi Wijeratne, declined to comment on the newspaper report. However, a UN spokesman in Colombo, Gordon Weiss, said the rebels refused permission to move the convoy.
"They (the Tigers) said it was unsafe for us to go forward," Weiss said, adding the UN estimated there were 200 patients to be evacuated.
Tens of thousands of people have been killed since the Tigers launched their war for a separate Tamil homeland in 1972.