The ICRC says thousands of people have been displaced
Tamil Tiger rebels say they have killed 51 soldiers near the town of Dharmapuram in northern Sri Lanka, a claim denied by the army.
It says the area was brought under government control two days ago and 20 rebels and seven soldiers were killed.
The army has already captured the rebel headquarters at Kilinochchi and the strategically important Elephant Pass.
The Jaffna peninsula and its capital have been regarded as the heart of the 25-year-old separatist insurgency.
Independent journalists are prevented by the government from travelling to the conflict zone, so it is impossible to verify the casualty claims made by both sides.
Meanwhile, Shivshankar Menon, India's Foreign Secretary, is meeting with Mahinda Rajapakse, the Sri Lankan president, as concern grows among India's Tamils.
More than 60 million Tamils live in the southern Indian state of Tamil Nadu.
They share close cultural and ethnic ties with Sri Lanka's Tamil population.
Political parties from Tamil Nadu, who form part of India's federal coalition, have called on the Indian government to intervene on behalf of the Tamils.
Officially, India favours a negotiated settlement, as many fear what a divided Sri Lanka could mean for India, says the BBC's Anbarasan Ethirajan in the Sri Lankan capital, Colombo.
The Indian government has not called for an end to the army's offensive in Sri Lanka but is offering more humanitarian aid.
The International Committee of the Red Cross (ICRC) says that the intense fighting in northern Sri Lanka has caused a "massive displacement" of civilians.
The Tigers have been fighting for a separate homeland for 25 years. At least 70,000 people have been killed in the insurgency.