Azerbaijan fears it may lose leverage over Armenia in the conflict over Nagorno-Karabakh if the border, closed since 1993, is reopened.
Armenian forces seized control of Nagorno-Karabakh and seven surrounding regions from Azerbaijan in the early 1990s, in a war that claimed an estimated 30,000 lives.
The two countries have cut direct economic and transport links and failed to negotiate a settlement on the region's status.
Agdam, where Thursday's reported violence occured, is one of seven Azeri districts surrounding Nagorno-Karabakh and is held by Armenian forces.
Despite a ceasefire line in and around Nagorno-Karabakh, clashes between forces on the frontline are common, with three Armenian soldiers killed by Azerbaijan forces in January.