The source could not confirm whether the assailant was the one arrested.
Following the shooting outside the church, the perpetrator fled, police said earlier. The motive for the attack remained unclear, the local authorities and the prosecutor's office said.
It is not yet clear why the Greek Orthodox shot the assailant in Lyon yesterday, the Mayor of Lyon, Gregory Doucet, said on TV.
In Lyon, a source close to the local prosecutors said the prosecutor's office had launched an investigation into the attempted assassination.
As reported by Reuters, police have launched a manhunt to locate the assailant who shot and injured 52-year-old priest Nikolaos Kakavelakis.
French Prime Minister Jean Castex said during his visit to Rouen and Saint-Étienne-du-Rouvray in north-western France that the government was determined to allow "everyone to practice their worship in complete safety and in complete freedom."
Our will is strong: our determination remains undiminished, it is France's honor, it is the republic's honor, "he said."
The incident comes three days after three people were killed by an armed Islamist, the perpetrator of the assault, in a stabbing attack in Notre-Dame de Nice.
Earlier, Charles Michel, the President of the European Council, condemned the attack in Lyon.
"Charles Michel said, adding that" all our thoughts are with the priest who is between life and death. "We condemn this new heinous act in Lyon."
"Freedom of conscience is guaranteed and must be respected in Europe and violence is intolerable and condemnable," said Charles Michel.