Armenia's Defense Ministry has said one of its troops was killed on the border with Azerbaijan overnight on June 18-19 but did not provide details.
Azerbaijani defense officials confirmed that shots had been fired in the area.
The Armenian side acknowledged that its forces had "fired with various-caliber firearms" at Azerbaijani military positions near their mutual border.
Relations between the Caucasus foes remain especially tense since they fought a six-week war in September-November 2020 over the territory of Nagorno-Karabakh and surrounding districts.
Skirmishes have broken out intermittently since heavy fighting over Nagorno-Karabakh following the breakup of the Soviet Union ended in an uneasy truce and "frozen conflict," with occasional deaths reported on both sides. Each side routinely blames the other for the violence.
Diplomatic initiatives to reestablish relations since the escalation of fighting more than a year ago have mostly stalled.
Nagorno-Karabakh, which had been under ethnic Armenian control for nearly three decades, is internationally recognized as part of Azerbaijan.
The 2020 fighting ended in a Russian-brokered cease-fire that returned large swaths of land to Azerbaijan, left around 2,000 Russian troops in the area to monitor the truce, and fed resentment in Yerevan.
Armenian Prime Minister Nikol Pashinian last week renewed his calls for the Russian-led Collective Security Treaty Organization (CSTO) to help his country deal with what he called serious security threats emanating from Azerbaijan.
Citing continuing "aggressive statements" by Baku, Pashinian said the military alliance of Russia, Armenia, and four other ex-Soviet states should specifically consider dispatching a monitoring mission to the Armenian-Azerbaijani border.
Article 2 of the CSTO's founding treaty prescribes a collective response to grave security threats facing member states.