Barbarin, the archbishop of Lyon in central France since 2002, has been accused of failing to remove a priest from his diocese when he became aware the man had sexually abused young boys 25 years ago.
Barbarin was being questioned by the police's family protection unit as part of a preliminary investigation, but has not been taken into custody, the source said.
State prosecutors will have to decide whether to pursue the case against Barbarin, which has shaken France's Catholic Church.
The scandal has raised questions over how deep Church
authorities should dig into abuse cases lurking in their past.
Barbarin's interrogation comes four days after Pope Francis issued a decree that senior Catholic officials guilty of negligence in child abuse cases can now be dismissed from office.
It was not immediately clear whether the papal decree would affect the Barbarin case.
The pope came under fire for meeting with Barbarin in May and has said it would be "nonsensical and imprudent" to seek the archbishop's resignation at this stage.
Barbarin has said he learned in 2007 that the priest, Bernard Preynat, had been accused of sexually abusing Scouts.
Preynat was only charged in January after a victim who was allegedly abused in the 1980s realised in 2015 that the priest was still in service. Several other victims have also come forward.
Barbarin has said that when he learned of the priest's past he immediately called a meeting with him and when he asked Preynat if he had committed further abuses since 1991 the priest swore he had not.
"You can reproach me for having believed him... but covering up means knowing and letting it happen," Barbarin said, adding he had "absolutely never" done that.