Britain also joined the European push for the talks, said diplomats, who added that a joint declaration could be issued at its conclusion—either by the entire council or its European members if a full consensus cannot be reached.
The clashes come in the middle of the annual UN General Assembly and raises fears of a war between ex-Soviet republics Azerbaijan and Armenia in the Caucasus.
Armenia and Azerbaijan have been locked in a territorial dispute over the ethnic Armenian region of Nagorny Karabakh for decades, with deadly fighting flaring up earlier this year and in 2016.
Increasingly deadly fighting raged Monday between Azerbaijani and Armenian forces, sparking bellicose rhetoric from Turkey despite international pleas for a halt in fighting between the longtime enemies.
Russia has a military alliance with Armenia, while Turkey backs Azerbaijan.
At least 95 people have been killed in the most recent fighting, including 11 civilians, according to the latest available tallies.