The prominent Catholic said some children and women among the persecuted Yazidi minority group who have fled sexual slavery at the hands of IS are missing out on the UK's Vulnerable Persons Relocation Scheme because they are not Syrian.
The Cardinal said: "The current approach means that many of the most vulnerable people, including Yezidi women and girls who have escaped from being held in sexual slavery by Daesh and cannot access appropriate support in the region, are excluded from this opportunity for resettlement abroad on the basis of their nationality."
In a statement released on the eve of the World Day for Migrants and Refugees, he wrote fleeing Iraqis and Syrians are involved in what the UK government already recognises as affectively the same conflict, adding "all refugees should be offered assistance".
With more Syrian refugees expected to be resettled in the UK this year - as part of a pledge to take in 20,000 people over five years - the Catholic Church has again said arriving families must be "welcomed and accommodated".
The Church said it has been "regularly" raising concern that not all of the most vulnerable refugees are able to access the relocation scheme.
It also called upon the UK government to continue pushing for peace across the region and provide the assistance necessary for rebuilding, "once peace is secured".
A Government spokesperson said: "We remain very concerned about the appalling crimes committed against Christians, Mandeans, Yazidis and other minorities, as well as the majority Muslim population, in Iraq and Syria.
"We are working closely with the Government of Iraq and announced £90 million of humanitarian assistance this year.
"The UK offers protection to Iraqi refugees who can qualify for resettlement programmes such as Gateway, Mandate and the Vulnerable Children's Resettlement Scheme. However only Syrian nationals qualify for the Syrian VPR."