The allegations concern Alois Schwarz, the current bishop of St. Poelten, but relate to his time as bishop of Gurk-Klagenfurt, in southern Austria, a post he held from 2001 until earlier this year.
On Tuesday high-ranking church officials in the Gurk-Klagenfurt diocese called a press conference where they laid out a series of allegations against Schwarz, principally of overseeing financial mismanagement that left the diocese with millions of euros in losses.
In addition, they said that Schwarz's relationship with a female director of a church educational institution had led to "gossip, rumour and speculation" and had clouded his judgement.
The officials said Schwarz's lifestyle had "left him open to blackmail as regards his requirement to be celibate".
Schwarz denies all wrongdoing, saying on Thursday that he welcomed the prospect of an investigation and would do his utmost to cooperate.
Responding specifically to the suspicion that he may not have stayed celibate, Schwarz said: "That's an allegation I won't put up with".
"I have always acted in accordance with the rules of the church," he told the ORF radio station.
The archbishop of Salzburg, Franz Lackner said he would begin the church's investigations in mid-January and that he hoped to "restore faith in the shepherds" among churchgoers.
According to local media reports, the last time the Austrian church was subject to such measures was in 2004, when a seminary was closed over allegations of widespread sexual misconduct, including possession of child pornography.