"The final critical act of law enforcement will be done in the coming days," Prime Minister Abiy Ahmed said.
Africa's youngest leader and the 2019 Nobel Peace Prize winner, Abiy launched air strikes and a ground offensive on Nov. 4 after accusing former comrades and the local ruling party, the Tigray People's Liberation Front (TPLF), of armed revolt.
Tigrayan leaders say Abiy, 44, who comes from Ethiopia's largest ethnic group the Oromo, has persecuted and purged them from government and security posts since taking office in 2018.
Tigrayan forces have fired rockets into the neighbouring nation of Eritrea, accusing Eritrean forces of helping the Ethiopian government side, an allegation denied by Asmara.
But the rocket fire has escalated a conflict which has already killed hundreds - one diplomatic source said thousands - and sent about 30,000 refugees into Sudan.
"How could they kill their own brothers and sisters? This is not our custom," Addis Ababa resident Fitawrari Million said of the Tigrayan leaders during a rally to honour federal soldiers.
The United Nations said a "full-scale humanitarian crisis" was unfolding. With communications largely down and media barred, Reuters could not independently verify assertions by either side nor the situation on the ground.
"People are coming out of Ethiopia really scared, afraid, with stories saying they have been fleeing heavy fighting and there's no sign of the fighting stopping," said Babar Baloch, spokesman for the UN High Commissioner for Refugees (UNHCR).
'BRING JUNTA TO JUSTICE'
Abiy's air force struck unspecified TPLF targets outside Tigray's capital Mekelle on Monday.
His government, in a statement by its special task force on Tigray, said on Tuesday federal forces had "liberated" the Raya, Chercher, Gugufto and Mehoni localities on the eastern front along with Shire on the western front of the conflict.
It had destroyed TPLF military bases in both areas and was now heading for the state capital Mekelle, the statement said. "The force of the junta is now retreating, and the army is marching to bring the TPLF junta to justice."
Tigray's leaders have accused federal forces of knocking out a dam and a sugar factory as well as "mercilessly" attacking people in the region of more than 5 million.
The government denies targeting civilians.
After an Amnesty International report of a massacre of scores of labourers, where it cited witnesses blaming TPLF forces, the UN rights chief said war crimes may have been committed.
The United States condemned the Tigrayans' weekend attack against the airport of Eritrea's capital, and urged a de-escalation of the war and protection of civilians.
In a sign the push towards Mekelle may not be going all to plan, TPLF leader Debretsion Gebremichael told Reuters battles were still ongoing around the southern town of Alamata that federal troops said they seized the previous day.
With hundreds of thousands of Tigrayans dependent on food aid even before the conflict, suffering is worsening even as humanitarian workers are pulling out for their own safety.
Two convoys moved about 600 people, mainly foreigners with international organisations, out of Tigray to the capital Addis Ababa this week, diplomats said.
Governments around Europe and Africa are clamouring for talks and even the Nobel committee voiced concern in rare comment on a past laureate's actions.
But Abiy has said he will only negotiate when rule of law is restored in Tigray, whose leaders are former military comrades whom he also partnered in government until 2018.
On Monday, his foreign minister went to Uganda and Kenya, while a Nigerian former president flew to Addis Ababa. Diplomats described a growing push for negotiations, but Ethiopia said it was simply explaining an internal conflict to outsiders.
Africa's second-most populous country has long been a powerful Western ally in a volatile region.
The fighting could jeopardise the recent opening up of Ethiopia's economy, stir ethnic bloodshed elsewhere around the vast nation of 115 million people, and tarnish the reputation of Abiy. He won his Nobel for pursuing peace with Eritrea and had drawn praise for opening up a repressive political system, though arrests and media curbs have tainted that of late.
His forces comprise around 140,000 personnel and are battle-hardened from fighting in Somalia, rebellious border regions and Eritrea in the past. But many officers were Tigrayan, much powerful weaponry is in Tigray, and the TPLF has seized the powerful Northern Command's headquarters in Mekelle.
The TPLF itself is a formidable rival also with a proud history: it spearheaded the toppling of a Marxist dictatorship in 1991 and bore the brunt of the 1998-2000 war with Eritrea.
Adding to the squeeze on the TPLF, state-affiliated Fana TV said Ethiopia had frozen bank accounts of 34 of its institutions including construction, trading, engineering, printing, electrical and bus companies.