In the statement on a "major humanitarian crisis unfolding," the ICRC said an estimated quarter of a million people were unable to flee from the 250 square kilometre (100 square mile) area and could not find shelter from the fighting.
The Colombo government as well as international rights groups have accused the Tamil Tigers of preventing civilians leaving the shrinking territory under their control.
Sri Lankan troops have dismantled a de facto state of the Tigers and say they have cornered the rebels in a narrow strip of jungle in the northeast.
"I stressed that military victories offer a political opportunity to restore life to normalcy in the northern province and throughout Sri Lanka," the Indian minister was quoted by the high commission as saying.
"We will work together with the government of Sri Lanka to enable all Sri Lankans, and particularly the Tamil community who have borne the brunt of the effects of the conflict, to lead normal lives as soon as possible."