Archbishop Leo Cushley made the offer when he gave his Thought for the Day on BBC Good Morning Scotland.
The invitation comes as the city council has announced that it will be taking in 100 Syrian refugees over the next 12 months while Glasgow City Council has pledged to house another 60 refugees on top of the 55 for which it already provides.
Archbishop Cushley said: “Many Scots Catholics – myself included – hail from families that once fled oppression and poverty in all sorts of places in the 19th and 20th century, to find a new home in this country.
“So how could we not place ourselves at the service of those 21st century newcomers who are in the greatest need?
“And, as a visible example of our willingness to help, I too would be happy to give a home to somebody fleeing war or oppression. Hopefully, though, we can find each family their own home. Scotland is a big country populated by big people with big hearts.
“Today, we ought to be able to welcome among us these people, real people, in real need, just as the Scots have always done in the past.”
He added that the crisis has been growing since the spring into the “greatest migration of peoples in Europe since the Second World War”.
Catholic Church bishops have also written to First Minister Nicola Sturgeon offering help to any refugees who arrive in Scotland.
The letter has been sent by Archbishop Philip Tartaglia on behalf of Scotland’s 452 parishes.
It pledges “complete cooperation” in giving both welcome and support to any refugees who arrive in Scotland in the months to come, offering to “place our parish resources at the service of those 21st century newcomers who find themselves in greatest need”.
Pope Francis, speaking last Sunday in St Peter’s Square in Rome, had invited every church, religious house and monastery in Europe to respond to the growing refugee crisis.
Pope Francis also announced that he will be opening the Vatican to two refugee families fleeing war and persecution.
Archbishop Cushley said the letter to the First Minister was part of the bishops’ response to that call.
Today thousands of people are expected to attend candelit vigils across Scotland to show solidarity for Syrian refugees and the victims of war.
Events will be held in Glasgow, Edinburgh, Dundee, Shetland and Orkney.
Archbishop Cushley is the Catholic Archbishop of St Andrews and Edinburgh and previously served as head of the English language section of the Vatican Secretariat of State.