Hundreds have died, more than 41,000 refugees have fled to Sudan, and there has been widespread destruction and uprooting of people from homes in the northern Tigray region.
The three-week war has spread to Eritrea, where the Tigrayans have fired rockets, and also affected Somalia where Ethiopia has disarmed several hundred Tigrayans in a peacekeeping force fighting al Qaeda-linked militants.
Reuters has been unable to verify statements made by either side since phone and internet connections to Tigray are down and access to the area is strictly controlled.
Prime Minister Abiy Ahmed's government said many Tigrayan combatants had responded to an ultimatum to lay down arms before a threatened offensive against Mekelle city, with half a million inhabitants. The deadline expires on Wednesday.
"Using the government's 72-hour period, a large number of Tigray militia and special forces are surrendering," a government taskforce said.
The battle-hardened Tigray People's Liberation Front (TPLF), which had ruled the region of more than 5 million people, gave a different version, saying their troops were keeping federal forces at bay and scoring some big victories.
Their spokesman Getachew Reda said an important army unit - which he termed the 21st mechanised division - was destroyed in an assault at Raya-Wajirat led by a former commander of that unit now fighting for the TPLF.
The prime minister's spokeswoman Billene Seyoum denied that.
The United States - which regards Ethiopia as a powerful ally in a turbulent region - France and Britain were the latest foreign powers to call for peace.
Washington backed African Union (AU) mediation efforts "to end this tragic conflict now", while Paris and London warned against ethnic discrimination.
The UN Security Council was to hold informal talks later on Tuesday over Tigray, according to a UN source and an email.
Abiy, who won the Nobel Peace Prize last year for ending a standoff with Eritrea, has said he will not negotiate with the TPLF though he does plan to receive AU envoys.
MYRIAD ETHNIC GROUPS
His predecessor, Prime Minister Hailemariam Desalegn, criticised international mediation efforts by "well-intentioned outsiders" in a piece for Foreign Policy magazine.
"The key problem in the international community's approach to Ethiopia is the assumption of moral equivalence, which leads foreign governments to adopt an attitude of false balance and bothsidesism" between the federal and Tigrayan sides, he said.
Abiy, whose parents are from the larger Oromo and Amhara groups, denies any ethnic overtones to his offensive, saying he is pursuing criminals who ambushed federal forces.
The TPLF says he wants to subdue Tigray to amass power.
Since taking office in 2018, the prime minister has removed many Tigrayans from government and security posts and arrested some on rights abuse and corruption charges, even though he was their former military comrade and coalition partner.
TPLF leader Debretsion Gebremichael has disputed the government version that Mekelle is encircled at a roughly 50km (31 mile) distance and told Reuters the ultimatum was a cover for government forces to regroup after defeats.
The conflict threatens to destabilise the vast nation of 115 million people from myriad ethnic groups whose struggles for greater resources and power intensified when Abiy took office.
In Geneva, the UN human rights chief voiced alarm over reports of tank and artillery build-ups outside Mekelle.
"We have seen an Ethiopian colonel come out and say there will be no mercy. On the other side you have had the TPLF leadership say they are ready to die," said Michelle Bachelet.
"This is the kind of rhetoric that is extremely worrying and that may provoke or may lead to serious violations of international humanitarian law."