Israel has said it will investigate the issue, but has not publicly acknowledged using the controversial chemical.
Al Jazeera has learnt that a total of 53 installations used by the United Nations Relief and Works agency, Unrwa, were damaged or destroyed during Israel's Gaza campaign including 37 schools - six of which are being used as emergency shelters - six health centres, and two warehouses.
White phosphorus - a high-incendiary substance that burns brightly and for long periods on contact with the air - is often used to produce smoke screens.
But it can also be used as a weapon producing extreme burns when it makes contact with human skin.
The Israeli newspaper Haaretz has reported a brigade of paratroop reservists fired about 20 white phosphorus shells into the built-up area of Beit Lahiya on January 17, which landed in the UN-run compound where the two Palestinian children were killed and severe burns were inflicted on 14 other people.
Amnesty International, the London-based rights group, has accused Israel of war crimes over its use of the munitions in heavily populated areas.
Children killed
International law forbids white phosphorus use against military targets within areas where civilians are concentrated, except when the targets are clearly separated and "all feasible precautions" are taken to avoid casualties among non-combatants.
If the claims are proved, Israel's use of the chemical could form the basis of war crimes charges.
Ban Ki-moon, the UN secretary-general, on Wednesday repeated his demand for a full explanation of Israel's attacks on UN facilities in the Gaza Strip.
With the UN secretary general unable to speak due to a sore throat, Lynn Pascoe, the UN under secretary-general for political affairs, read out a statement on his behalf saying Ban wanted "a thorough investigation by Israel into every single one of these incidents".
The attacks, which Ban described as "outrageous", included strikes on a compound of the UN agency providing aid for Palestinian refugees (Unrwa) and on a UN school last week during Israel's three-week war on Gaza.
"I expect a full explanation of each incident and that those responsible will be held accountable for their actions," the statement quoted Ban as saying.
Deadly weapons
The Israeli military has also been accused of using Dense Inert Metal Explosive (Dime) weapons in urban areas, causing horrific abdominal and leg injuries.
When detonated, a Dime device expels a blade of charged tungsten dust that burns and destroys everything within a four-metre radius.
Israel has been criticised by human rights groups and foreign officials over its suspected use of a number of weapons during its aerial, naval and ground assault on the Palestinian territory in which over 1,300 Palestinians were killed.
The UN's International Atomic Energy Agency (IAEA) said on Wednesday it will open an investigation into whether Israel used depleted uranium, which is added to munitions as its density allows them to penetrate armour more easily, during the conflict.
It is thought that the dust left at blast sites after the weapons have hit also pose a health risk, but a definitive link has not yet been proven.