Residents in Amhara’s regional capital, Bahir Dar, reported hearing heavy gunfire.
A toxic political atmosphere
By Emmanuel Igunza, BBC Africa, Nairobi
These are tumultuous time for Ethiopia and Prime Minister Abiy, who is already facing increased ethnic tensions.
Chief of Staff Seare Mekonnen had only served as head of the military for one year having being appointed by Mr Abiy, who made sweeping changes in the security apparatus when he took office last April.
It is clear that the prime minister still has significant opposition from within the military who are opposed to his style of leadership.
The killing of Amhara’s governor is also a big blow for Mr Abiy, who is credited with installing Ambachew Mekonnen in office.
He was a key ally in Amhara, which is itself facing security problems and clamour from some groups for greater autonomy from the central government.
The first general election since Mr Abiy came into office is supposed to be held next year but it is very hard to see how this will continue in a country that is highly polarised. The atmosphere is just too toxic.
Why is Amhara so important?
The homeland of the Amhara ethnic group is the country’s second most populous region and has given Ethiopia its state language, Amharic.
Violence between the Amhara and Gumuz ethnic groups left dozens of people dead last month in Amhara and its neighbouring region, Benishangul Gumuz.
Ethnic violence, typically sparked by land disputes, has displaced nearly three million people across Ethiopia.
Another issue the prime minister is having to grapple with is unrest within the military.
In October, he said that hundreds of soldiers who had marched to his office to demand a pay rise had wanted to kill him.
Mr Abiy survived a grenade attack at a rally in June last year which killed two people and left more than 100 injured.
Who is the alleged organiser of the coup attempt?
Asaminew Tsige was among a group of high-ranking military officers released from prison early last year when the previous government moved to free political detainees in response to public pressure.
The general had been in custody for nine years for allegedly plotting a coup.