Mr Fischer, who served as Australia's first resident ambassador to the Holy See between 2008 and 2012, recommended the position as an under-appreciated but important post that presented an opportunity for the government to use Mr Abbott's high-level experience.

"It should not be underestimated and deals with everything from the SKA [Square Kilometre Array] radio telescope to helping bomb victims in the 'pope's hospital' but above all else is a key listening post including to the Balkans and eastern Europe," Mr Fischer said.

"May I humbly suggest that in many ways the capabilities of former prime minister Tony Abbott could be very usefully employed for all Australians by sending him to Rome. Indeed, other nations such as Poland, have sent former prime ministers as ambassadors to the Vatican."

In his concession speech on Saturday, Mr Abbott said he expected to continue in public life in some way.

The Vatican-based role is being openly speculated about in some Coalition circles as an option for Mr Abbott. Speculation has also focused on other higher profile diplomatic posts in Washington and London as well as potential roles based in Australia.

Australia has had formal diplomatic relations with the Holy See since 1973. Mr Fischer became the first resident ambassador in 2008.

The current ambassador Melissa Hitchman, a career diplomat, is approaching the end of her three-year posting.

Mr Fischer said it be useful to have a someone like Mr Abbott keeping tabs on the Catholic Church as Pope Francis heads towards an "inevitable resignation". He said he had not talked to Mr Abbott about the idea.

Mr Abbott, a devout Catholic, studied to become a priest in the 1980s before turning to journalism and politics.

He has been vocal on matters of faith and is close to George Pell, who was one of the most senior figures at the Vatican until his recent conviction on five counts of child sexual abuse. Abbott has defended Pell - who is appealing - as a "fine man" and called him after the verdict.

Mr Fischer has added his voice to a chorus of Abbott's conservative backers calling on the government to appoint the former prime minister to an important public role. Businessman Roger Corbett, Liberal senator Eric Abetz and Liberal MP Craig Kelly have all expressed a hope that Mr Abbott would be offered a role.

A spokesman for Mr Abbott did not respond to a request for comment. A spokesman for Foreign Minister Marise Payne declined to comment.