Troops have been moving north after capturing Kilinochchi
The Sri Lankan government says troops have reached the southern outskirts of the strategic Elephant Pass, which is the key link to the Jaffna peninsula.
Troops have been pushing north since capturing the Tamil Tigers' de facto capital of Kilinochchi on Friday.
There has been no comment yet from the rebels, who have held control of the pass since 2000.
Tamil sources said the rebels killed 53 soldiers at the weekend. The army denies such losses.
There is no independent confirmation of casualties and both sides regularly overstate the numbers that they inflict.
Tamil Tiger rebels have been fighting for a separate homeland for the island's ethnic Tamil minority for the past 25 years. At least 70,000 people have been killed in the conflict.
Military spokesman Udaya Nanayakkara said troops had gained control of the southern outskirts of Elephant Pass.
He said soldiers had overpowered the resistance of the rebels there and that operations were ongoing to "bring the entire area under our control".
The website of Sri Lanka's defence ministry says troops have taken control of the village of Kurinchattiv and have entered Thamilamadam, both of which lie just to the south of the causeway.
The loss of Elephant Pass would be another major blow to the Tamil Tigers, correspondents say, leaving Mullaitivu as their sole remaining major base.
Capturing Elephant Pass would allow government troops in the peninsula to link up with the mainland along the key A9 road.
Further south, soldiers also captured the strategic town of Oddusuddan at the weekend, the army said on Monday. Oddusudan sits at an important crossroads to Mullaitivu.
The pro-Tamil website TamilNet says the rebels killed 53 soldiers and injured 80 at the weekend in fighting on the main road towards Mullaitivu.
Military officials denied that troops had suffered heavy casualties and said the army had recovered the bodies of 12 Tamil Tigers.
Maj Gen Jagath Dias, who led the capture of Kilinochchi, said he was confident of taking the rebels' remaining strongholds in the north and east.
"Day by day, the Tigers' territory is shrinking and their numbers are dwindling," he was quoted by the Associated Press agency as saying.
"The objective of finishing this war won't be that long off," said Gen Dias.
The BBC's Roland Buerk was among a number of reporters taken by the army to Kilinochchi following its capture.
Much of the town was in ruins, he said, with little sign of normal life - just 20 or so of the town's population remained.
Troops were gearing up for operations further east. Attack helicopters flew overhead and every few minutes there was an artillery barrage, our correspondent added.