Eritrea denies TPLF allegations it has sent soldiers over the border to back Abiy's offensive against the Tigrayan forces, who are also an old foe of Eritrea's.
The Ethiopian government has repeatedly said it will not engage in talks with the Tigray People's Liberation Front (TPLF), which it regards as a renegade administration, pointing to what the government says was a surprise attack the group launched on federal troops in Dansha, sparking the conflict. On Saturday it said Adigrat had also fallen, about 116 km (72 miles) north of Mekelle.
The army of Abiy and Eritrean President Isaias Afwerki "inflicted heavy casualties on innocent civilians in Adigrat", the TPLF's communications bureau said in a statement on Facebook.
According to Reuters, it's hard to verify the current positions of the forces, as internet and phone lines have been down since the conflict began.
The fighting in Ethiopia's north erupted in early November, when the federal government accused the ruling party in the rebel Tigray region of attacking a local military base.
Hundreds, possibly thousands, have died and more than 30,000 refugees have fled to Sudan.
In a statement Saturday, Abiy praised his army's advance. They said in a broadcast on Friday they were making progress on the southern and northern fronts.
"Together with the rest of Ethiopia, we will work to ensure that all humanitarian needs will be addressed", Abiy added.
The Tigray region has been facing a rebellion since earlier this month.
The TPLF is popular in its home region and dominated national politics from 1991 until Abiy took office. "Citizens of the cities and towns under Federal command in the Tigray region are already easing into the regular routine with the protection of the Federal Police", it reads.
TPLF leaders have complained of being sidelined and blamed for the country's woes.
The envoys are Joaquim Chissano, former president of Mozambique; Ellen Johnson-Sirleaf, former president of Liberia; and Kgalema Motlanthe, former president of South Africa, who will travel to Ethiopia in "coming days".
U.N. Secretary-General Antonio Guterres issued a statement late Friday, welcoming the appointments and commended Ramaphosa "for this initiative" and extended the U.N.'s full support "for ensuring a peaceful, stable and prosperous Ethiopia".
The government's task force on Tigray tweeted, "News circulating that the envoys will be travelling to Ethiopia to mediate between the Federal Government and TPLF's criminal element is fake".