Meanwhile, Ethiopia's human rights commission says it is sending investigators to Mai Kadra in the northern Tigray region, where Amnesty International had reported mass killings amid the conflict in the area.
Ethiopia has said that forces loyal to the ruling party in the northern Tigray region had fired into neighbouring Amhara region, raising fears that ongoing fighting could draw in other parts of the country.
"In the late hours of November 13, 2020, a rocket was fired towards Bahir Dar & Gondar cities. As a result, the airport areas have sustained damages," a government statement said.
"The TPLF junta is utilising the last of the weaponry within its arsenals," the statement said, referring to the Tigray People's Liberation Front, the Tigray ruling party.
TPLF leader Debretsion Gebremichael told AFP Saturday he did not have information about the reported attacks but noted that TPLF leaders have said "any airport used to attack Tigray will be a legitimate target".
The airports in both Bahir Dar, the regional capital, and in Gondar are used by both military and civilian aircraft.
Ethiopia's human rights commission has said it is sending a team of investigators to Mai Kadra in the northern Tigray region, where Amnesty International reported mass killings amid a military offensive.
The commission also said in a statement late on Friday that there was a reasonable risk of "ethnic profiling" in connection with the conflict, which pits Prime Minister Abiy Ahmed's troops against local forces.
Abiy Ahmed, winner of last year's Nobel Peace Prize, ordered military operations in Tigray last week, shocking the international community which fears the start of a long and bloody civil war.
Abiy says the operation is in response to attacks on two federal military camps by Tigrayans, who once dominated Ethiopian politics and claim they have been sidelined and targeted under Abiy.
The United Nations rights chief Michelle Bachelet on Friday warned of possible war crimes in Tigray, while the UN's special adviser on the prevention of genocide Pramila Patten "condemned reports of targeted attacks against civilians based on their ethnicity or religion".
READ MORE: UN calls for full inquiry into possible war crimes in Ethiopia
Fleeing Ethiopians describe horror of fighting
Barefoot and nearing collapse, 10-year-old Tediest Gezriel is among the thousands of Ethiopians streaming into Sudan fleeing intense fighting, reporting artillery bombardments and air-strikes in their homeland of Tigray.
Tediest, starving after walking 30 kilometres barefoot for two-days through the baking heat, was separated from his family in the fear and the chaos.
With a communication blackout in Ethiopia's northern Tigray region, the accounts by the exhausted and terrified refugees pouring across the remote frontier into neighbouring Sudan provide some of the first eyewitness accounts of the week-old conflict.
"They bombed with artillery, and the air force raided," said Asmara Tefsay, a 31-year old mother. "Then we saw the soldiers approaching and I fled with my two children, my mother and my father."
Many of the refugees appear traumatised by the sheer intensity of the bombardments they say were carried out by the Ethiopian army.
"I saw women giving birth on the road, but then continuing to walk because they feared the Ethiopian soldiers would kill them," said Roni Gezergil, a female engineer aged 25.
READ MORE: Sudan reopens famine camps to shelter thousands of Ethiopian refugees
Hid in well
Gabera Solasi, a 22-year-old mathematics student at university, took the only shelter he could find when the heavy bombardments grew close.
"I hid in a well during the bombing, and then fled during a lull," Gabera said.
"Now I am in Sudan and I think the war will continue. I am not sure I will ever be able to go back to university."
Tediest, wearing the orange T-shirt and grey trousers he had on when he ran from the fighting, approaches the Su danese soldiers and a few aid workers, holding his hand out pleading for food.
He's not the only one begging for scraps of bread; there are hundreds of children in the streets.
Some are with their parents, others, like Tediest, were separated in the panic as they followed the long column of thousands of others heading for Sudan.
Alsir Khaled, regional head of Sudan's refugee agency, said that as of Friday evening, at least 21,000 Ethiopians had crossed into eastern Sudan seeking help.
"They keep on coming," he told AFP, adding that many arrived from the Ethiopian town of Humera, where some of the heaviest fighting has been reported.
READ MORE: Amnesty: Scores of civilians killed in 'massacre' in Ethiopia's Tigray
Little government support
Journalists who toured the area said there was little support for the Ethiopians from the state or charities, with aid coming mostly from villagers.
"During the two-day walk, I only drank water, there was nothing to eat," said Tsefay Salomon, a 23-year-old student.
"Once we crossed the border, some Sudanese took us in a car to this village. The local community gave us food, but it is so little that we keep it especially for children."
Sudanese villagers offer what help they can, but some fear the thousands arriving from Ethiopia will strain their already very limited resources.
"Instead of being in camps, many have settled in our fields," said Sudanese farmer Jamal Adam.
"They have cut our trees to protect themselves from the sun, while others are sleeping under the stars in our fields. It is now time for the sorghum harvest, and I risk losing everything."
But for Ethiopians, many who fear they have lost their homes and livelihoods, they say they had no other choice but to seek sanctuary in Sudan.
"It is a war against the Tigrayan people," Roni Gezergil said.
"It is not a political war. It is a war to exterminate us."