Tensions have blown up along the Line of Control, the de facto border in Kashmir, over the past week with the two incidents drawing angry protests from the two sides who have disputed control of the Himalayan region for more than six decades.
India has said Pakistan soldiers beheaded an Indian soldier during a clash on Tuesday in which two troops were killed. Pakistan said one of its soldiers was killed first on Sunday.
"On the January 8 alleged incident, no official complaint has been received from either the Pakistani army or Indian army," Nesirky said.
"UNMOGIP is aware that the Pakistani army and Indian army are in contact via the hot line and urges both sides to respect the ceasefire and de-escalate tensions through dialogue."
He added that Pakistan's military had made an official complaint about the January 6 clash and the UN observers "will conduct an investigation as soon as possible in accordance with its mandate."
There are about 40 UN military observers in Kashmir monitoring a ceasefire that was officially agreed in 1949. Kashmir is the cause of two of three wars fought by the rival neighbors since their independence from Britain in 1947.
Pakistan is serving a two-year term on the UN Security Council, over which it is presiding for January. India has just finished a two-year stint on the council.