Fr Angelo Chizzolini, from Lugiria in northwest Italy, made the comment after Pope Francis recently called the Catholic Church to take in as many refugees as it could in response to the current crisis.
According to the Times, Mr Chizzolini has previously offered refuge to non-EU migrants in the nearby area of Brescia, but they left the property in bad condition.
The Catholic Church in England and Wales has officially declared it will help local councils re-settle Syrian refugees when they come to Britain, while the Christian charity Home for Good has said more than 8,000 people have approached it with the desire of adopting an unaccompanied Syrian child refugee.
Clarifying his comments to a local newspaper, he said: "I said that I would not host refugees in my home.
"As a private citizen I have a home in Brescia and I can do what I like with that, and yes, I said that rather than opening the door to a migrant, I would set fire to it."
Matteo Salvini, the leader of Italy's far-right Northern League party, has expressed his support for Fr Chizzolini's actions, posting "I'm with Father Angelo" on his Facebook page.