Ethiopians fleeing war in the northern Tigray region have crossed a border river into neighbouring Sudan, some in boats but some swimming or wading through the water.

At least three rockets were fired at Eritrea's capital from Ethiopia, diplomats said, as the deadly fighting in Tigray appeared to spill across an international border and bring some observers' worst fears to life.

Speaking to Reuters in the Sudanese border town of Hamdayat, refugees gave accounts of the escalating conflict in Tigray state, where Government forces are battling fighters loyal to rebellious local leaders.

The small town is home to a camp hosting 8,000 refugees. Several hundred arrived on Saturday morning with hundreds more crowded onto the rocky bank of the Tekeze River.

Refugees told stories of artillery attacks and shooting in the streets, with fighting spilling over into neighbouring Amhara state.

"We are hungry and we're afraid that they will kill us," said an old woman, referring to government troops battling the Tigray People's Liberation Front (TPLF).

Refugees said one person drowned while trying to cross the night before.

Niqisti, 42, said her brother was shot dead by Government-allied militiamen in front of their home in Humera in Tigray state and her small restaurant was looted.

Sudanese officials were registering refugees.

"People are hungry and the flow of refugees is continuing but we have little to offer," said Salah Ramadan, head of the border town's administration.

Rockets aimed at Eritrea

Hundreds of people have been killed since Ethiopian Prime Minister Abiy Ahmed sent the national defence force on an offensive against local troops in Tigray last week, accusing them of attacking federal troops.

Mr Abiy, who won a Nobel Peace Prize for a 2018 peace pact with Eritrea, has rejected growing calls by the United States and others for an immediate de-escalation.

At least three rockets appeared to be aimed at the airport in Eritrea's capital, Asmara, hours after the Tigray regional government warned it might attack.

It has accused Eritrea of attacking it at the invitation of Ethiopia's federal government since the conflict in northern Ethiopia erupted on November 4.

Tigray's leader, Debretsion Gebremichael, said on Tuesday that Eritrea had sent troops across the border in support of Ethiopian government forces but provided no evidence.

Eritrea's Foreign Minister Osman Saleh Mohammed denied this.

"We were shelled by artillery volleys from across the Eritrean border," said Naksiam Guru, a 22-year refugee who lives near the border.

Burhani Abraham, 31, who arrived four days ago with his wife and three-year-old child, also reported shelling from Eritrea.

"I'm very hungry," he said, sitting in the crowded makeshift camp.

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'Afraid of war and death'

Experts have warned that Eritrea, long at bitter odds with the TPLF, could be pulled into Ethiopia's growing conflict that has killed untold hundreds of people on each side and sent some 25,000 refugees fleeing into Sudan.

A 26-year-old farmer from the Tigray region said the TPLF had tried to recruit him but he ran away. He did not want to give his name.

"I'm a simple farmer growing corn … I'm afraid of war and death," he said.

Fears of ethnic targeting are rising.

Ethiopia's Human Rights Commission, appointed by the government but independent, said it was sending a team of investigators to the Tigray town of Mai Kadra to investigate reports of mass killings.

The rights group Amnesty International said on Thursday that scores and possibly hundreds of civilians were stabbed and hacked to death in the region on November 9, citing images and witnesses.

It said it had not been able to independently confirm who was responsible, but said the witnesses had blamed fighters loyal to Tigray's leaders.

The Tigray state government denied any involvement by TPLF members or the state's special police force in "this most tragic event."

The international community is warning against deadly ethnic tensions.

The UN office on genocide prevention has said the rhetoric sets a "dangerous trajectory that heightens the risk of genocide, war crimes, ethnic cleansing and crimes against humanity".

Desperate families cannot reach relatives, and the United Nations and other humanitarian organisations warn of disaster as food, fuel and other supplies run short for millions of people.

"The military escalation in Ethiopia is risking the stability of the whole country and wider region," the European Union's commissioner for crisis management, Janez Lenarcic, tweeted.

There is no sign of any easing in the fighting.

AP/Reuters