Armenian and Azerbaijani forces engaged in new fighting on Tuesday over the Nagorno-Karabakh region despite pleas to observe a ceasefire, as the Red Cross warned hundreds of thousands were already affected by the conflict.

More than two weeks of fighting between the Caucasus rivals has left almost 600 dead, including 73 civilians, according to a tally based on partial tolls from both sides.

The Nagorno-Karabakh region of Azerbaijan, overwhelmingly populated by Armenians, has been controlled by Armenians since a 1990s war that erupted as the Soviet Union fell. 

But Azerbaijan has never hidden its desire to win back control and no state has ever recognised Nagorno-Karabakh's declaration of independence.

The fighting has been the most intense since a 1994 ceasefire ended the initial post-Soviet war. Even a humanitarian truce to allow exchanges of prisoners and dead has been too much to implement.

"Civilians are dying or suffering life-changing injuries," said International Committee of the Red Cross Eurasia regional director Martin Schuepp in a statement.

"Homes, businesses and once-busy streets are being reduced to rubble."

He said hundreds of thousands of people across the region were affected, with healthcare services coming under strain and even attacked in some cases.

The Nagorno-Karabakh separatist authorities accused Azerbaijan of launching an offensive in the south, north and northeast of the region. 

Baku accused Armenians of launching strikes on the districts of Goranboy, Terter and Agdam - Azerbaijani territory outside Nagorno-Karabakh.

An AFP correspondent in Terter close to the front line reported hearing heavy shelling nearby and seeing a rocket launcher drive by as Azerbaijani forces sought to fire up into the mountains.

The daily fighting has made a mockery of the ceasefire agreed between the Armenian and Azerbaijani foreign ministers in the early hours of Saturday in Moscow after 11 hours of talks.

Russian Foreign Minister Sergei Lavrov on Monday acknowledged that the ceasefire was not holding and urged both sides to stop fighting.

The European Union and Iran have also called on the parties to properly observe the ceasefire.

The search for a long-term solution to the conflict, one of the most enduring headaches left after the fall of the Soviet Union, is in the hands of the Minsk Group of regional powers chaired by France, Russia and the US.

But both sides are heavily entrenched in their views and three decades of diplomacy has brought little progress.