(Recasts with U.N. statement)

NAIROBI, Dec 27 (Reuters) - The United Nations said on Thursday it was investigating an outbreak of gunfire on the volatile border between Eritrea and Ethiopia.

A statement from the U.N. peacekeeping mission UNMEE charged with monitoring the border -- where 70,000 people were killed in a 1998-2000 war -- said observers on the Eritrean side heard shots in the early hours of Wednesday.

"UNMEE expresses concern about this firing incident between Ethiopia and Eritrea and is calling on both parties to show maximum restraint," it said.

"The force is in the process of investigating the incident."

Eritrea's government accused Ethiopia of attacking its security forces, describing it as part of ongoing provocation along the still disputed border.

In a statement on its Web site shabait.com, Asmara said the relatively small-scale raid, which it said was on Tuesday, targeted its troops and allied militias in the South Tsorona region, inside a former buffer zone, but failed.

"(The) attack comes in continuation to (Ethiopia's) ongoing provocation and aggression in the Gash-Barka and Southern regions, whereby it planted mines, carried out incursions, abducted nationals and burned crop fields to the ground," the Eritrean statement said.

An Ethiopian official rejected the report and said Addis Ababa had no reason to provoke a new conflict with Asmara.

"If any country is war-mongering, it is Eritrea," Bereket Simon, special adviser to Ethiopian Prime Minister Meles Zenawi, told Reuters. "Ethiopia, at present, is focusing on its economic development, ensuring democracy and good governance internally."

Ethiopian officials routinely reject Eritrea's version of border incidents. Bereket said there was a clash in Gash-Barka this week, but it was "purely an Eritrean internal affair".

"It is well known there are a number of opposition groups waging war internally against the (Eritrean) regime," he said.

Earlier this month, U.S. Secretary of State Condoleezza Rice urged Ethiopia to avoid raising tensions with Eritrea.

In November, an international commission charged with setting the 1,000-km (620-mile) frontier dissolved itself, leaving the two states to work it out alone. (Additional reporting by Tsegaye Tadesse in Addis Ababa; Writing by Daniel Wallis and Andrew Cawthorne)

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