Officials in Sudan, which shares a border with the Tigray region, say at least 6,000 Ethiopians have arrived in the country.
The Sudanese authorities are preparing for a further 200,000 people to cross in the coming days.
Al Sir Khalid, a commissioner for refugees in Sudan’s Kassala state, told the BBC’s Focus on Africa programme that some of those fleeing had walked for two or three days to escape “bombardment”, and that they included Ethiopian soldiers.
What’s it like in Tigray at the moment?
The federal government has cut off the phone lines and internet, Alula Solomon, the director of a Tigray media house based in the US told BBC Newsday.
His family, who live in Tigray the border near Sudan, were able to contact him by using Sudanese telephone lines. They have now fled to Sudan.
Are other countries involved in the conflict?
On Tuesday, Mr Debretsion accused the neighbouring Eritrean government of launching attacks on the Tigrayan state.
The accusation was denied by both the Eritrean government and the Ethiopian national army.
Mr Abiy won the 2019 Nobel Peace Prize for ending a two-decade conflict with Eritrea, which began when the TPLF dominated the Ethiopian government. Relations between the TPLF and Eritrea remain extremely bad.
What is behind the conflict?
The TPLF was the most powerful bit of Ethiopia’s ruling coalition for many years but Mr Abiy curbed its influence after winning power in 2018 on the back of anti-government protests.
Relations got worse last year after Mr Abiy dissolved the ruling coalition, which was made up of several ethnically based regional parties.
He merged them into a single, national party, the Prosperity Party, which the TPLF refused to join.
Tigrayan leaders say they have been unfairly targeted by purges and allegations of corruption, and say Mr Abiy is an illegitimate leader, because his mandate ran out when he postponed national elections due to coronavirus.
The simmering row boiled over in September after the TPLF defied the nationwide ban on elections, and held a vote which was declared illegal by the central government.
Why did Mr Abiy win the Nobel Peace Prize?
Mr Abiy won the prize in 2019 for his role in resolving the border conflict with Eritrea.
Ethiopia and Eritrea fought a bitter border war from 1998-2000, which killed tens of thousands of people.
Although a ceasefire was signed in 2000, the neighbours technically remained at war until July 2018, when Mr Abiy and Eritrea’s President Isaias Afwerki signed a peace deal.