The United Nations refugee agency said on Sunday that more than 20,000 people have crossed into Sudan from Ethiopia’s northern Tigray region, where federal government troops are battling forces loyal to the Tigray People’s Liberation Front (TPLF), the party of the regional government.
Sudanese state media put the number of refugees at almost 25,000.
The 12-day conflict between Ethiopia's government and forces in the country's Tigray region has now escalated.
On Saturday, the Tigrayan forces fired rockets across the border into neighboring Eritrea, after claiming Ethiopian soldiers were using an Eritrean airport to attack Tigray. No casualties have been reported yet.
However, a day after the attack, Ethiopia's prime minister appeared to deny the accusations in a tweet.
Abiy said that Ethiopia was "more than capable of attaining the objectives of the operation by itself".
Tigrayan leader Debretsion Gebremichael has said that his forces have been fighting 16 divisions of the Eritrean army on several fronts for the past few days.
The Tigrayans have also accused Eritrean forces of crossing into Ethiopia to back federal forces there.
But while Eritrea and Ethiopia have denied co-operating in the conflict, experts say that reports of fighting along the border, and of Ethiopian soldiers being treated in Eritrean hospitals, suggest the opposite is true.
Analysts argue that with missiles being fired into Eritrea, the conflict has now become far more serious and is likely to be harder to stop. The consequences are expected to cause instability in the whole region.
But many people familiar with recent political developments had predicted trouble was brewing when Ethiopia's Prime Minister Abiy Ahmed became a strong ally of Eritrea's leader, Isaias Afwerki.
Now, analysts say, the two men have a common enemy - the Tigrayan politicians of the TPLF who dominated Ethiopia's political scene for years, including the time when Ethiopia and Eritrea fought a border war that left tens of thousands dead.