The main damage to UN property came on January 15 when Israeli shells hit a UN compound, setting fire to a warehouse and destroying badly-needed food and medical aid.
UN officials say they have evidence that white phosphorous, a smokescreen agent that can cause severe burns, was used in the attack that left three people injured.
Israel said Hamas fighters had used the compound to launch attacks on its forces but later apologised for the incident.
A UN inquiry last year put the cost of damage to seven UN buildings in Gaza during the 22-day assault at $11.2m, almost all of it caused by Israeli forces.
Loss adjusters hired by the UN subsequently reduced that by $750,000, Nesirky said.
Nesirky said the UN's compensation claims related both to the property damage and to minor injuries suffered by 11 employees.
Israel's onslaught on Gaza killed more than 1,400 Palestinians and left much of the territory in ruins. A total of 13 Israelis were killed during the conflict.