Abiy, the victor of last year's Nobel Prize, late Sunday gave the Tigray People's Liberation Front (TPLF) 72 hours to surrender - an ultimatum rejected by the leader of the dissident northern region, who has said his people are "ready to die" for their homeland.
Religious and historical sites, institutions and residential areas would not be targeted, he said.
Ethiopia's army - which in recent days said it was advancing on Mekele with tanks - had been directed "to conclude the third and final phase" against the TPLF, Abiy said.
In the "final phase" of the military offensive that PM Abiy launched following the expiry of the 72-hour ultimatum, he spoke of a military strategy in which top personnel of the TPLF would be brought to justice without harming civilians or properties in and around Mekelle. He called on residents of the capital of the northern region to stay at home and away from military targets. It denied government claims several of its troops had already surrendered. But there were multiple reports on Thursday that the refugee exodus was being blocked by Ethiopian soldiers, who were stationed near the Sudan border.
Tigray remains under a communications blackout and media access to the region has been restricted, making independent verification of claims from both sides hard. Internet, mobile phones and landlines are all down, making it hard to contact those accused in the preliminary report.
Ethiopian refugees fleeing from the ongoing fighting in Tigray region, queue for water, at the Fashaga camp, on the Sudan-Ethiopia border, in Kassala state, Sudan, November 24, 2020. The TPLF is heavily armed, with an estimated 250,000 fighters.
Ethiopia says the matter is an internal affair affecting its national security. "Next to the casualties, the danger of a major humanitarian crisis is imminent", European Union commissioner for crisis management Janez Lenarcic tweeted.
But a full-scale attack on the city of more than 300,000 people could lead to the deaths of thousands in punishing urban warfare.
The finance minister cited an alleged November 9 massacre in the town of Mai-Kadre in the region, where the Ethiopian Human Rights Commission (EHRC) says that a local youth group helped kill hundreds of civilians, charges that followed an Amnesty International report that "scores, and likely hundreds" were killed.
Earlier this week, his military warned that it would show "no mercy" in its assault on Mekele if the city's residents refused to flee. The Ethiopian government has countered by reporting surrenders among TPLF forces. "They need to ensure the safety of civilians under their control", it said.
The war in Tigray erupted on November 4 after rising tensions between the Ethiopian government and the TPLF, a long-dominant faction that lost political power when Mr. Abiy became prime minister in 2018.
All that changes when Abiy Ahmed becomes prime minister in April 2018, the first ever from the Oromo ethnic group, the country's largest.
When Mr Abiy postponed a national election because of coronavirus in June, relations further deteriorated.
In September the party held its own election, which the central government said was "illegal".