Azerbaijan's defence ministry said on Saturday that Armenian forces had fired on its troops overnight, and that Azerbaijan army units took "retaliatory measures".
It said Armenian units opened small arms fire on Azerbaijani soldiers in Sadarak, north of Nakhchivan, an exclave of Azerbaijan that borders Armenia, Turkey and Iran.
The ministry's statement did not say if there had been any casualties.
Reuters could not independently verify the reported incident, which came against the background of rising tensions between Armenia and Azerbaijan in recent days.
The Armenian government and state media said Armenian Prime Minister Nikol Pashinyan held phone conversations on Saturday with the leaders of France, Germany and Iran and with US Secretary of State Antony Blinken.
The government said Pashinyan told Blinken and Iranian President Ebrahim Raisi that tensions were rising on the border and Azerbaijan was concentrating troops around the Armenian-populated enclave of Nagorno-Karabakh. Baku has denied this.
The government said Pashinyan told Blinken and Raisi he was ready to hold an urgent meeting with Azerbaijani President Ilham Aliyev to reduce tensions.
State news agency Armenpress said Pashinyan had similar conversations with French President Emmanuel Macron and German Chancellor Olaf Scholz.
Azerbaijan meanwhile denounced the holding on Saturday of a presidential election in Nagorno-Karabakh, a territory that is internationally recognised as part of Azerbaijan but is populated by about 120,000 ethnic Armenians.
Nagorno-Karabakh established de facto independence in a war in the early 1990s after the collapse of the Soviet Union, but Azerbaijan recaptured significant amounts of territory in its most recent war with Armenia in 2020.
In a statement, Azerbaijan's foreign ministry called the ethnic Armenian leadership of Karabakh a "puppet separatist regime" and said the vote was illegal.
"The Republic of Azerbaijan will resolutely counter threats to its sovereignty and territorial integrity within its internationally recognised borders," the statement said.
"The only way to achieve peace and stability in the region is the unconditional and complete withdrawal of the Armenian armed forces from the Karabakh region of Azerbaijan and the disbandment of the puppet regime."