About 100,000 civilians are estimated to be in Fallujah which, in January 2014, became the first Iraqi city to be captured by the self-proclaimed Islamic State group.
The city's population was three times bigger before the war.
Now Iraq has launched an assault to retake the militant stronghold.
Residents in the city, 50 kilometres west of the capital Baghdad, have reported sporadic shelling around the city centre, as forces continue to shell IS targets.
International concern is mounting for the security of civilians who have limited access to food, water and healthcare and now at risk of being used as human shields.
Rupert Colville, from the United Nations Office of the High Commissioner for Human Rights, is urging Iraq to protect civilians in the town.
"We also urge the government of Iraq to ensure that the protection of civilians is paramount in its offensive on Fallujah, and we call on all parties to adhere strictly to international humanitarian law, including the principles of distinction and proportionality in the conduct of hostilities."
With reports of I-S squads set up in Fallujah with orders to kill anyone trying to flee or surrender, the International Committee of the Red Cross says it's also concerned about the well-being and safety of civilians in Fallujah.
Ralph El Hage, from the ICRC in Baghdad, says the agency has been unable to provide any assistance to communities affected by the violence.
"We have not been able to enter the city for over a year and a half now, even more, and that is due to the inability to reach sort of a humanitarian agreement with all parties in the armed conflict currently ongoing in Iraq, to allow the safe entrance of our humanitarian staff, to give us the security guarantees that nothing will happen to our staff so we are able to enter Fallujah."
The ICRC also says it's seeking to negotiate safe access to Fallujah to deliver humanitarian assistance.
Prime Minister Haider al-Abadi has told Iraqi TV the armed forces have been "instructed to preserve the lives of citizens in Fallujah and protect public and private property".
But Raje Barakat, from Anbar Provincial Council, says some Fallujah residents have been killed in the latest fighting.
"Islamic State fighters try to hide in civilian areas and some civilians were either martyred or wounded. We try to distance some of the families but IS is using them as human shields."
Efforts to recapture Fallujah from IS is being conducted by the army, police, counter-terrorism forces, local tribal fighters and a coalition of Shi'ite Muslim militias known as Hashid Shaabi.
Prominent Hashid Shaabi leader Hadi Al-Amiri says the group supports the Iraqi army in the battle.
"We are encircling the city but we will not enter it. This is a policy, which we adopted in our operations in all areas. We said that we will enter only if there is a need and a necessity. We entered Al-Dhuluiya when the people of Al-Dhuluiya people said that they could not do it without us and this was the case in Tikrit and Baiji. Our decision is to encircle the city only. We will only go in if the Iraqi forces are in absolute need of support in Falluja. We took the decision of not entering Fallujah."
The United States military is estimating IS forces in the city number between 500 and 700.
Other sources say IS numbers in Fallujah could be in the low thousands.
