A "full-scale humanitarian crisis" is unfolding in Ethiopia, the United Nations refugee agency has said, with more than 27,000 people now having fled heavy fighting to Sudan.
Babar Baloch, spokesman of the UN High Commissioner for Refugees, also told a Geneva news briefing: "UNHCR is on stand-by to provide assistance in Tigray when access and security allow."
He said some 4,000 people are fleeing across the border every day.
"Refugees fleeing the fighting continue to arrive exhausted from the long trek to safety, with few belongings," Mr Baloch said. "The pace is very rapid."
He stressed that Sudan was now experiencing "an influx unseen over the last two decades in this part of the country."
Ethiopia's prime minister has warned that a deadline for rebel northern forces to lay down arms had expired, paving the way for a final push on the Tigray region's capital in a two-week conflict destabilising the Horn of Africa.
Africa's youngest leader and the 2019 Nobel Peace Prize winner, Abiy Ahmed launched air strikes and a ground offensive on 4 November after accusing the local ruling party, the Tigray People's Liberation Front (TPLF), of armed revolt.
The TPLF dominated Ethiopian politics for three decades before Mr Abiy came to power in 2018.
A bitter feud has grown as the party became sidelined, growing ever more defiant towards the central government.
Tigrayan leaders say Mr Abiy, who comes from the largest ethnic group the Oromo, has persecuted and purged them from government and security positions since taking office.
Tigrayan forces fired rockets into the neighbouring nation of Eritrea this weekend, widening a conflict that has already killed hundreds of combatants and civilians, with one diplomatic source saying it could be thousands.
US Secretary of State Mike Pompeo condemned the action, urging the TPLF and the Ethiopian authorities to take immediate steps to de-escalate the conflict.
Posting on Twitter, he called for a restoration of peace and the protection of civilians.
Mr Abiy's warning came after his forces struck unspecified TPLF targets outside the Tigray capital Mekelle, a government emergency taskforce said.
With hundreds of thousands of Tigrayans dependent on food aid even before the conflict, suffering is worsening fast even as humanitarian workers are scaling back for security reasons.
UN humanitarian agency spokesperson Jens Laerke said given the number of people crossing the border there may be "massive displacement" inside Tigray.
Mr Baloch also cautioned that the conflict was also "a major ongoing concern for the Eritrean refugee population" in Tigray, which he said numbered nearly 100,000.
"Potential for further displacement of refugees inside the country is increasingly a real possibility," he said, warning that the "humanitarian situation as result of this crisis is growing rapidly."
A convoy of four buses and several cars, carrying about 400 foreigners from Mekelle, was expected to arrive in the capital Addis Ababa tomorrow, five diplomatic sources said.
Another convoy of about 200 people, mainly workers for international organisations, reached the capital yesterday.
The United Nations and governments around Europe and Africa are pressing for talks and even the Nobel committee expressed deep concern and called for peace in a rare comment on the actions of a past laureate.
But Mr Abiy has resisted, saying he will only negotiate when rule of law is restored in Tigray.
The fighting could jeopardise the recent opening up of Ethiopia's economy, stir ethnic bloodshed elsewhere around the vast nation of 115 million people.