Fighting that started on Friday and continued through the weekend has cut the main road connecting government-held regions of the peninsula with the mainland, a military official said.
The defence ministry has said 36 troops and 150 Tamil rebels have been killed in the fighting on the peninsula and near Jaffna city which the rebels controlled as their capital from 1990 to 1995.
The main airfield of Palaly on the peninsula has also been shut down, with private airlines ordered to halt flights there, after taking several artillery hits on Friday, officials said.
"There is a day time curfew in Jaffna in addition to the night curfew in the high security zones," a military official said.
"But we have reports of civilians moving out of their homes and sheltering in public buildings.
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The bulk of supplies and troops to Jaffna are sent by sea from the northeast port of Trincomalee which also came under artillery attacks on Saturday.
"The army pulled back from some of the defensive