The pontiff made the announcement after calling for Catholic parishes, convents and monasteries across Europe to do the same.
He said refugees who are "fleeing death" should be offered a home and cited Mother Teresa, the European-born nun who cared for the poorest in India, in making his appeal in remarks to pilgrims and tourists in St Peter's Square.
"Faced with the tragedy of tens of thousands of refugees who are fleeing death by war and by hunger, and who are on a path toward a hope for life, the Gospel calls us to be neighbours to the smallest and most abandoned, to give them concrete hope," Francis said.
It is not enough to say "have courage, hang in there," he added.
"May every parish, every religious community, every monastery, every sanctuary in Europe host a family, starting with my diocese of Rome," Francis said.
He also asked bishops throughout Europe to have their dioceses take up his call to "express the Gospel in concrete terms and take in a family of refugees".
There are two churches in the tiny city-state. Francis said both would welcome a refugee family but the Vatican has not released any further details.
Vatican spokesman, Revd Ciro Benedettini, quoted the pope's chief alms-giver as saying the Vatican is now deciding which families will be hosted.
Meanwhile David Cameron is to set out details of the Government's plans to re-settle thousands of refugees.
Officials have been working on the scheme throughout the weekend after the Prime Minister dramatically dropped his opposition to taking more people seeking sanctuary from the conflict.
Mr Cameron - who will give full details in a Commons statement - was effectively forced to act after harrowing photographs of the body of three-year-old Aylan Kurdi washed up on a Mediterranean beach galvanised public opinion.
He had previously argued taking in refugees would simply encourage more people to risk the hazardous sea crossing to Europe which has already claimed thousands of lives.
Under the proposal, announced on Friday during a visit to Portugal and Spain, Britain will take refugees directly from the camps in countries bordering Syria - avoiding the need for them to put themselves in the hands of people traffickers.
It represents a major expansion of an existing Government programme to provide places for the most vulnerable refugees which has so far seen 216 Syrians re-settled in Britain.
Critics have said the plan will do nothing to help the tens of thousands who have already made it across to Europe and are seeking to make their way to the wealthier northern nations, most notably Germany, to claim asylum.
However Mr Cameron has been adamant Britain will not join a proposed EU scheme to redistribute some 160,000 people among the member states, despite risking alienating key allies like Germany's Chancellor Angela Merkel.
Ministers refused to be drawn over the weekend on how many people will be admitted to the UK, although the Government is likely to want to appear as generous as possible prompting speculation that it could be up to 10,000.