The UN's refugee agency says more than 30,000 people have fled to Sudan because of the ongoing conflict in Ethiopia's Tigray region.
UNHCR emergency coordinator Jens Hesemann described the situation at the border to BBC Newsday.
"The situation is increasing everyday, yesterday alone we had 2,300 new arrivals, the day before we had over 5,000 new arrivals," he said.
The agency's transit centre has been overwhelmed as they were meant to accommodate fewer refugees.
"The community has been generous by sharing the little they have with the arriving refugees, the authorities have also been very good about it they are identifying land where we can set these refugees for the inland which is very important," Mr Hesemann said.
The refugees are concerned about their safety.
"They speak of violence some of them have said they have fled direct fighting. They are really worried and what is significant is people looking over their shoulders, asking are we safe here," Mr Hesemann said.