The priest, who has Greek nationality, was closing his church when the attack happened and is now in a serious condition, said the source, who asked not to be named.
The interior ministry tweeted that “an incident is underway” in the southeastern city of Lyon.
“Security and emergency personnel are at the scene,” the ministry added, warning people to “avoid the area” where the attack took place.
The shooting comes three days after three people were killed in a knife rampage inside a church in the southern town of Nice.
A Tunisian suspect was shot by police near the scene of that attack.
France was already on edge after the republication in early September of cartoons of the prophet Mohammed by the Charlie Hebdo weekly, which was followed by an attack outside its former offices, the beheading of a teacher, and the attack in Nice.
Foreign Minister Jean-Yves Le Drian has warned that French citizens face a security risk “wherever they are”, saying alerts had been sent to all French nationals abroad.
France went into a second coronavirus lockdown on Friday but the government has exempted places of worship until Monday, allowing them to celebrate the Christian All Saints' Day on Sunday.
France has been on high alert since the January 2015 massacre at the satirical weekly magazine Charlie Hebdo marked the beginning of a wave of jihadist attacks that have killed more than 250 people.
Tensions have heightened since last month, when the trial opened for 14 suspected accomplices in that attack.
After the deadly attack in Nice, President Emmanuel Macron announced increased surveillance of churches by France's on-the-street military force, which is to be bolstered to 7,000 troops from 3,000.
Security at schools will also be boosted, he said. Schools are remaining open during the new lockdown.