Millions of people are said to be running out of food and medicines.
UN officials told AFP there was still no access on Friday, despite a deal allowing “unimpeded” humanitarian access to government-controlled areas.
The military entered the regional capital Mekelle last weekend and said the month-long conflict with forces of the TPLF group was over.
Hundreds of people have reportedly been killed in the fighting with the Tigray People’s Liberation Front. Tens of thousands of people have been displaced, with a wave of refugees fleeing into Sudan.
The BBC managed to speak on Thursday to people in Mekelle who said “fighting is still going on in places near the city”. The TPLF says it is still fighting.
Ethiopia’s minister in charge of democratisation, Zadig Abraha, dismissed this, saying “there is no war”.
Getachew Reda, a member of the TPLF Executive Committee, told the BBC the situation in Mekelle was “very tense”.
“People have no appetite whatsoever to countenance the invading forces,” he said.
TPLF troops had withdrawn from Mekelle to spare the city a massive bombardment from the attacking forces, he said, but fighting was taking place on the city’s outskirts.
“My forces are fighting for the self-determination rights of their people. They are fighting gallantly and heroically and there is no reason why they should surrender,” he told the BBC’s Focus on Africa programme.
However he said the TPLF was ready to negotiate.
“While we would not submit to any arm-twisting, we remain beholden to the need for peace,” he said.