In a statement posted on Twitter Sunday, Abiy said that government forces were now "in the third and final phase" of their military operation in the area.
"We urge you to surrender peacefully within the next 72 hours, recognizing that you are at a point of no return," he added, addressing members of the local ruling party, the Tigray People's Liberation Front (TPLF), directly.
Ethiopian federal forces have been at war with the regional government of Tigray
, which borders Eritrea and Sudan, since the beginning of November, when Ethiopian Prime Minister Abiy Ahmed announced a military offensive including air strikes.
The conflict followed tension over Tigray's unilateral decision to elect a regional administration against Abiy's wishes.
"Our operation aims to end the impunity that has prevailed for far too long and hold accountable individuals and groups under the laws of the land," Abiy said at the time.
Hundreds have died and at least 30,000 refugees have fled to neigboring Sudan, according to the United Nations.
Tigrayan leaders have accused Abiy's federal forces of killing innocent civilians while targeting churches and homes. The government has denied targeting civilians and CNN has been unable to verify claims from either parties due to the communications blackout. Internet, mobile phones and landlines are all down.
In his statement on Sunday, Abiy called on the people of Mekelle to "play a key role" in the defeat of the TPLF by "standing in solidarity with the national defence."