Civilians who managed to flee besieged Fallujah have reported cases of starvation in the Iraqi city that government forces are trying to recapture from Islamic State militants, the Norwegian Refugee Council said on Thursday.

"If they stay in Fallujah they face possible starvation, if they try to escape they risk being killed getting out," NRC media coordinator Becky Bakr Abdulla said in a report, citing refugee accounts.

The Iraqi army launched an offensive on Monday to dislodge the ultra-hardline Sunni militants from Fallujah, 50 km (32 miles) west of Baghdad. Falluja was the first Iraqi city to fall under Islamic State control, in January 2014, and has been under a tight siege for about six months.