The Kilinto prison on the outskirts of the capital, Addis Ababa, is where many opposition figures and journalists are held. Activists first raised concerns about the fire on Saturday.
Fana Broadcasting Corporate cited a government statement as saying 21 inmates died from stampede and suffocation. The other two were killed while trying to escape.
The government originally said only one person died in the fire.
The latest government statement gives no details on how the fire began. It says an investigation has begun.
The Oromia and Amhara regions have held protests in recent weeks that residents and rights groups say have left dozens dead. Regional concerns that sparked the demonstrations have broadened into anti-government protests.
Ethiopia's government, a close security ally of the West, is often accused of silencing dissent.
Recent video obtained by The Associated Press showed Ethiopian security forces beating, kicking and dragging several protesters during a rare demonstration in the capital.
Ambassador Samantha Power called the recent violence in Ethiopia "extremely serious" and called for a transparent and independent investigation.
She said the US has asked the government to allow people to protest peacefully.
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