A panel of experts recommended the measure, the source added, declining to say whether the clerics had already been named since the diocese's predator priest scandal came to light in March, rocking France's Catholic Church.
French judicial authorities are aware of all four dossiers, the source said.
The dismissals come three weeks after Cardinal Philippe Barbarin, the archbishop of Lyon, was questioned by police over allegations that he covered up the sexual abuse of boy scouts.
Preynat was relieved of his duties last year.
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Pope Francis issued a decree early in June that senior Catholic officials guilty of negligence in child abuse cases can now be dismissed from office.
The nine-member expert panel -- which included a psychiatrist, a psychoanalyst, a doctor and Church legal expert -- also recommended that the diocese keep an eye on other priests who have already been investigated.
Barbarin had said in April that he would reveal the panel's conclusions by the end of June.
He said the members were tasked with "studying and analysing the cases of certain priests whose situation is problematic in terms of their pastoral activity, civil or Church law in the affective and sexual domain."
The panel, which also includes a lay woman and man, both of whom are parents, has met seven times.