Emergency services were sent to St Simon’s Church in Partick Bridge Street – the parish priest there is Canon Peter McBride, who is from Renton – after the alarm was raised at 02:40.
More than 30 firefighters are fighting the blaze and the Scottish Fire and Rescue service say one person, believed to be Sister Mary, a nun who lives next door to the church, was given precautionary treatment at the scene.
The Archdiocese of Glasgow said it was a “sad end for a much-loved” church.
A spokeswoman for the Scottish Fire and Rescue Service (SFRS) said six fire appliances were dispatched and crews were expected to remain at the site for some time.
Pictures and videos posted on social media show flames and smoke billowing from the Roman Catholic church.
Nearby residents were advised to keep windows and doors closed.
Photos of the church taken at first light show significant fire damage to the roof and front windows.
A spokesman for the Archdiocese of Glasgow said the cause of the blaze was “not immediately apparent” but they would work with SFRS to investigate.
He added: “The destruction of St Simon’s will be a blow to people far beyond the west end of Glasgow.
“Though small in size, St Simon’s was well-frequented and was the spiritual home of the Polish community in the west of Scotland who had established a shrine there.
“St Simon’s was a much-loved landmark.”
Built in 1858, St Simon’s is the third oldest Catholic church in the Archdiocese of Glasgow, after Glasgow Cathedral and St Mary’s church in the city’s east end.
The church’s links to Glasgow Polish community date back to World War Two.
It was used by Polish soldiers based at nearby Yorkhill Barracks. A plaque in front of the church commemorates this connection.
Weekly masses continue to be said in Polish by a priest from the chaplaincy at the Sikorski Polish Club in Glasgow.
The club said in a statement: “Extremely sad news this morning. Saint Simon’s Church was at the heart of our Polish Community with Polish services every week. It will be greatly missed.”