Mr Abiy said that Chief of Staff General Seare Mekonnen had been attacked by "mercenaries".
Earlier, prime minister Abiy Ahmed said his government had foiled a coup attempt in the Amhara region.
BBC understands that in Amhara itself, regional governor Ambachew Mekonnen was killed along with an adviser.
"Chief of Mission personnel are advised to shelter in place", the Embassy said on its website.
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The situation in the Amhara - located in northwestern part of Ethiopia and is one of nine regional states in the country - is now under full control of the Federal Government, the statement added.
The US Embassy in Addis Ababa issued a security alert saying it was aware of shots being fired in the capital, as well as violence in and around Bahir Dar.
A statement from his office did not give details on who was believed responsible for the attack, the latest blow to his efforts to stabilise and reform the Horn of Africa nation.
Earlier, PM Ahmed announced the government had put down a coup attempt in the autonomous Amhara state in the country's north. Amhara in Ethiopia's northern highlands, is the homeland of the ethnic group by the same name, and is the birthplace of many of its emperors as well as the national language Amharic.
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Alors que l'annulation de sa victoire de mars a suscité indignation et colère parmi ses partisans, M. Dans une allocution après la publication des premiers résultats le donnant vainqueur, M.
State media named him as General Asamnew Tsige, who was Amhara's head of security.
But the premier's shake-up of the military and intelligence services has earned him powerful enemies, while his government is struggling to rein in powerful figures in Ethiopia's myriad ethnic groups fighting the federal government and each other for greater influence and resources.
Since coming to power in April past year, Abiy has pushed to open up the once isolated, security-obsessed country.
Abiy came to power in April 2018 after two years of anti-government protests which began in the Oromia region but also swept Amhara - whose people make up about 30 per cent of the population and are the second largest ethnic group after the Oromo. He also established relations with neighboring Eritrea.
Observers say that Abiy's plans to hold an election in 2020 has stirred up resentment in local politics, with other regional parties contesting the hold of those within the EPRDF, and seen a rise in ethno-nationalism.
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"A similar attempt was committed past year in June but we successfully overcame it", Abiy said in reference to a grenade attack at a rally he attended in 2018.