Azerbaijan's Defence Ministry said three of its troops and seven Armenian troops were killed in "heavy" clashes.
The Armenian authorities controlling Nagorno-Karabakh denied those losses and said "more than 10" Azerbaijani troops died or were wounded on Sunday.
Nagorno-Karabakh has been in Armenian hands since a war that ended in 1994.
A shaky ceasefire has been in place since then, but it is regularly broken by shelling - leaving hundreds of thousands of war refugees still displaced in both republics.
Internationally the mountainous territory is still considered part of Azerbaijan.
Heavy shelling also took place on Friday, in which four Armenian troops and an Azerbaijani soldier were reported killed.
Later the foreign ministers of both countries - Azerbaijan's Elmar Mammadyarov and Armenia's Edvard Nalbandian - met in New York, in the presence of mediators from the Organisation for Security and Co-operation in Europe (OSCE). However, no breakthrough was reported.
Casualties have been reported in renewed shelling between Armenian and Azerbaijani troops in the disputed Caucasus region of Nagorno-Karabakh.