Ethiopia has not commented on the reported fighting in the Tsorona area, about half-way along the frontier.
Residents on the Ethiopian side of the border reported hearing gunfire and seeing a large movement of troops and artillery towards the border.
A peace deal in 2000 ended the countries' two-year war but it has not been fully implemented.
Ever since, the countries have been in a state of "neither war nor peace", says the BBC's Ethiopia correspondent Emmanuel Igunza.
Eritrea says the tense relations with Ethiopia are why it has national conscription, which can last for decades.
It's not clear why the fighting has erupted now as neither country has issued any reasons.
But in recent months both sides have upped the rhetoric with the latest verbal salvo coming from Eritrea's President Isaias Afwerki during celebrations to mark 25 years of the country's independence.
He accused Ethiopia of being hostile to Eritrea's sovereignty.
Earlier this year, Ethiopia's Prime Minister Hailemariam Desalegn said his country was ready to take "proportionate military action against Eritrea" for what he described as "continuous acts of provocation and destabilisation of Ethiopia".
The conflict, over the exact location of the border led to the deaths of an estimated 80,000 people.
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Ethiopia "unleashed an attack against Eritrea on the Tsorona Central Front. The purpose and ramifications of this attack are not clear," the Eritrean government said in a short statement issued on Sunday night.
It does not mention any casualties.
Ethiopia's Information Minister Getachew Reda, who is out of the country, told the BBC he was not aware of the fighting.
As part of the Algiers peace agreement signed in 2000 both countries agreed to accept the ruling of an independent boundary commission over the location of the frontier as "final and binding".
But after the commission ruled that the disputed town of Badme, where the conflict began, was in Eritrea, Ethiopia at first refused to agree to the border demarcation and then called for dialogue before it would implement the decision.
This was rejected by Eritrea and there has been an impasse ever since, although clashes have been rare.
Eritrea has accused Ethiopia of launching an attack at the countries' heavily-militarised border.