According to Breitbart News and The Trumpet, Cardinal Christoph Schonborn made the comment in a discussion with Lutheran Bishop Michael Bünker on Austrian television.
The Archbishop went on to say that he was becoming "more cautious" about how to handle the world refugee crisis and that his home country was "worried" about the levels of migration to Europe.
He said: "We cannot accommodate all refugees. We must first see that they can find and live in their homeland again.
"We have had to learn, this [crisis] goes well beyond our capacity and ability."
It's thought two million refugees have applied for asylum in Europe since last year.
Many of them have been accepted by Germany - something which has divided the opinions of voters there.
The German far-right has seen more support in local elections after hundreds of robberies and assaults were reported in the German city of Cologne on News Year's Eve 2015. The majority were carried out by non-Germans.
Germany has also seen a string of terrorist attacks in the last two years, the most recent of which saw a truck plow into revellers at a Christmas market in Berlin, killing twelve people.
Others have praised German Chancellor Angela Merkel's policy of welcoming migrants, highlighting the war, persecution and poverty they are fleeing - some of it caused or made worse by Western policies.
Bishop Michael Bunker said in the discussion with Cardinal Schonborn: "Now it is very strongly claimed: refugees bring terrorism. At the same time, one must also say that terrorism brings refugees.
"Most people on the run are fleeing precisely from this kind of terrorism, which we now have experienced in Nice and Berlin."
