A German Catholic bishop has flown to Rome to meet Vatican officials after being criticised for huge cost overruns on a luxury residence and allegedly lying under oath.

A spokesman confirmed that Limburg Bishop Franz-Peter Tebartz-van Elst had departed but would not say when, or how long he would be away.

He declined to comment on media reports the prelate flew on a budget airline.

Bishop Tebartz-van Elst has caused a crisis in the German church after building a luxury residence and office complex at a time when Pope Francis is stressing humility and service to the poor.

"The bishop has made it clear that any decision about his service as a bishop lies in the hands of the Holy Father [Pope Francis]," said a statement issued by the diocese on Saturday.

"The bishop is saddened by the escalation of the current discussion.

"He sees and regrets that many believers are suffering under the current situation," it added.

An initial audit of his spending, ordered after a Vatican monitor visited Limburg last month, revealed the project cost at least €31m.

Bishop Tebartz-van Elst has also been accused of lying under oath about a first-class flight to visit poverty programmes in India.

The head of the German Church, Archbishop Robert Zollitsch, has said the scandal was hurting the whole church.

He said he would discuss it with Pope Francis during a visit to Rome this week.

The Pope's response will be closely watched as a barometer of how far he will go to promote frugality and simplicity in a church plagued for decades by scandals of clerical sexual abuse and opaque financial transactions at the Vatican bank.

The Limburg case presents special problems because Bishop Tebartz-van Elst, at 53, is too young to simply be retired off.

In the Catholic Church, a bishop retains his status even if he loses his office.

State prosecutors in Hamburg said last week they wanted the bishop to be fined for making false affidavits about the first-class flight to India.

Bishop Tebartz-van Elst said he flew business class.

However, Der Spiegel magazine has made public a mobile phone video recording of a conversation that triggered action by prosecutors.