He said his agency has met with some refugees in the Ethiopian capital, Addis Ababa, and he again urged unhindered humanitarian access to Tigray.
But Tigray residents have asserted that gunfire came from the direction of Eritrea as the conflict began.
It said the refugee camps have had no access to food or other supplies for more than a month.
It was not clear where, but the group also supports the Eritrean refugees.
About 1 million are now thought to be displaced.
Around midday, the Associated Press saw more than a dozen people waiting on the Ethiopian side of the border.
People continue to flee Ethiopia several days after prime minister Abiy Ahmed declared victory in the conflict, and reports of fighting continue in the Tigray region, which remains largely cut off from the world. Refugees have recounted horrific journeys of fleeing attacks and arriving on foot after two or three days of walking in the heat.
They are not doing anything for us.
The world has its laws.
People are dying from hunger, by guns, they are dying.
Why are they being silent?
Nearly 1 million have been displaced by the fighting.
Each regards the other as illegal in a power struggle that has been months in the making.