Not all Christians will return to Mosul after it is liberated from Islamic State, according to a Chaldean priest ministering to refugees from the city.
The operation to drive the ultra-hardline militants from Mosul began in October and has recaptured villages and towns surrounding the city, and most of Mosul's eastern half.
New tactics and better coordination have helped Iraqi forces advance faster since they launched a new phase of the operation more than 10 days ago.
Iraqi forces have fought their way into two more southeastern districts but their advances are being slowed by Islamic State's tactic of using civilians for cover.
However, Fr Thabit Mekko, currently living in Erbil with many of his congregation who fled the city, told Fides news agency that it was "too early to think about a return of Christians who have fled from their homes. Such a case will be considered only when security is assured," he said.
Mekko said: "Many families have not yet decided what they will do. Not all those who left Mosul in front of the advance of Daesh will return."
The situation in Mosul, the last major city in Iraq held by Islamic State, remains desperately hard for civilians.
A spokesman for the rapid response units of Iraq's federal police Lieutenant-Colonel Abdel Amir al-Mohammedawi, told Reuters IS was hiding among civilians.
He said rapid response units and Iraqi army units had fought their way into the Palestine and Sumer districts in the last day, but that IS fighters were firing at civilians trying to flee.
"The families, when they see Iraqi forces coming, flee from the areas controlled by Daesh towards the Iraqi forces, holding up white flags, and Daesh bomb them with mortars and Molotov cocktails, and also shoot at them.
"Whenever they (IS) withdraw from a district, they shell it at random, and it's heavy shelling," he said.
