Patriarch Sako has told European Union representatives unless the current conflict in the country is brought under control "Christians will be left with just a symbolic presence".
Hundreds of Iraqi Christians have been forced to flee their homes since Islamic militant group ISIS started taking control of some towns and cities in the country.
"We are known to be a disinterested mediator seeking the good of the country.
"Where fighting groups refuse to meet outside, when we invite them to our Churches to talk, they come." Said Patriarch Sako.
He was meeting with an EU delegation in Brussels at a meeting organised by Catholic charity Aid to the Church in Need.
Syrian Catholic Archbishop Yohanna Petros Mouche of Mosul and Chaldean Catholic Archbishop Youssif Mirkis of Kirkuk were also at the meeting where they met EU Council President Herman Van Rompuy.
Archbishop Mirkis told the politicians: "There's so much panic that few Christians see their future in Iraq."
European People's Party MEP Tunne Kelam said the current Iraq crisis has made EU politicians more aware of the fate of Christians in the Middle East.
He said: "We cannot remain indifferent to their situation. The EU should do its utmost to assist them and create conditions that Christians, the oldest known inhabitants of that region, can remain there in conditions of equality and mutual respect."
Since the fall of Sadam Hussain after the US led invasion of 2003 the number of Christians in Iraq has fallen from more than 1.5 million to 400,000.
Patriarch Sako said: "Under Saddam we had security but no religious freedom. Today we have religious freedom but no security".
EU heads of State will discuss the Iraq crisis on 16th July.