By Saif Hameed

AMIRIYAT FALLUJA, Iraq (Reuters) - Tens of thousands of civilians stranded inside Islamic State-controlled Falluja are at risk of disease outbreaks as Iraqi government forces press their assault to retake the city, a World Health Organization (WHO) official said on Thursday.

Islamic State has tightened control over civilian movement in central Falluja, where an estimated 40,000 people are stuck with little water or food, as commandos from an elite counter-terrorism force inch closer to the city's main government building more than three weeks after the offensive began.

The ultra-hardline militants, who by U.S. estimates have lost almost half of the northern and western territory they seized when Iraqi forces partially collapsed in 2014, have used residents as human shields to slow the military's advance and thwart an international air campaign backing them.

Falluja, an hour's drive west of Baghdad, is seen as a launchpad for Islamic State bombings in the capital, making the offensive a crucial part of the government's campaign to improve security, although U.S. allies would prefer to concentrate on IS-held Mosul, Iraq's second largest city in the far north.

Children inside Falluja have not received immunizations since the militants took control in 2014, said Ala Alwan, WHO's regional director for the eastern Mediterranean, who visited displacement camps in Amiriyat Falluja, a town southeast of Falluja.

"The low level of immunity coupled with poor hygiene conditions raises the risk of disease outbreaks, such as measles," he said in a statement, and hundreds of pregnant women were also in urgent need of reproductive health services.

Poor sanitation could also spread infectious diseases like cholera and skin diseases in addition to exacerbating chronic illnesses, Alwan said during a field visit where some displaced people complained about a shortage of electricity.

Iraq suffered a major cholera outbreak last year with more than 1,800 cases including six deaths.

Aid groups providing food, water and other supplies to escapees lack access to the city itself. Falluja was besieged by government forces for around six months before the current advance began, prompting the United Nations and rights groups to warn of an imminent humanitarian crisis.

The number of people displaced from Falluja has reached 53,000, according to U.N. statistics. Alwan said they also struggled to access health services due to inadequate funds.

"The major problem is a lack of funding, life-saving funding that is absolutely required to prevent disease outbreaks," he told reporters during his visit.

The humanitarian community this year requested $861 million to assist 7.3 million Iraqis in need across the country, but so far it has only received 31 percent of this amount.

WHO has set up a healthcare center and mobile clinics to serve displaced people, and is operating disease early-warning sites across Anbar province, where Falluja is located, to detect and respond to possible outbreaks, Alwan said.

"We have huge concerns about communicable diseases and preventing epidemics, but the good news is that we have no major outbreaks," he added.

Iraq said on Monday it had made arrests as it investigates allegations that Shi'ite Muslim militiamen helping the army recover Falluja had executed dozens of Sunni Muslim men fleeing the city, whose population is mainly Sunni.

The role of militias in the battle of Falluja alongside the Iraqi army had already raised fears of sectarian killings.

Falluja is a historic bastion of the Sunni insurgency against U.S. forces that toppled dictator Saddam Hussein, a Sunni, in 2003, and the Shi'ite-led governments that followed.

Enemies of Islamic State have launched major offensives against the jihadists on other fronts, including a push by U.S.-backed forces against the city of Manbij in northern Syria.

They amount to the most sustained pressure on IS since it proclaimed a religious caliphate in 2014.

(Additional reporting and writing by Stephen Kalin; Editing by Mark Heinrich)