As Julia Gillard and Tony Abbott vie for the backing of key independent MPs to form Australia's first minority government in 70 years, questions have been raised over whether Ms Bryce should excuse herself from adjudicating over who will get the job.
The extraordinary prospect of the Governor-General not being able to perform her crucial role in the event of a political deadlock was confirmed yesterday when she released a statement acknowledging concerns about her relationship to Mr Shorten, who is married to her daughter Chloe.
''The Governor-General is seeking advice on concerns raised about her personal position in the current political circumstances,'' a Government House spokeswoman said.
In a further statement last night, her office revealed that the Commonwealth Solicitor-General, Stephen Gageler, SC, had agreed to provide advice on the issue.
If Ms Bryce concluded that her relationship with Mr Shorten could be perceived as a conflict of interest, the task of resolving a political deadlock might fall to the longest-serving state governor, Marie Bashir in New South Wales.
Mr Shorten, one of the architects of former prime minister Kevin Rudd's demise and a potential Labor leadership candidate, refused to comment yesterday.
But he addressed the issue on ABC's Insiders on Sunday. ''If it does come to that sort of constitutional crisis … your mother-in-law, the Governor-General, would have to make the call,'' host Barrie Cassidy said.
''In terms of the Governor-General's role, she will carry out her role, I'm sure,'' Mr Shorten replied. ''In terms of any other point that you may be humorously alluding to, I'm not going to impugn that office.''
The power to appoint a prime minister if an election results in a hung Parliament is among the so-called ''reserve powers'' of the Governor-General.
Constitutional law experts were divided yesterday on whether, in the event of a political deadlock, there was a potential conflict for Ms Bryce.
Professor George Williams said Ms Bryce had done the right thing in taking advice. ''It's good that she's acted early in a proactive way to get it looked at. It's something that I think is sensible to get an opinion on,'' he told The Age.
''If everything runs as it should, no conflict issue can even arise because it will be Parliament that will make a decision and that will be the end of the matter.
''But, if things do go awry and extraordinary things happen and she finds herself in the middle of a political maelstrom, then in that case I think there is a real possibility that perceptions are strong enough that it would just be wise not to provide a distraction and to ensure that any decision is made without even the slightest suggestion of a conflict.''
Professor Greg Craven of the Australian Catholic University said Ms Bryce's family connection to Mr Shorten could not lead to any reasonable perception of bias.
''We've had governors-general before who have been former ministers of governments over which they have had to preside - that's a far closer connection than this sort of thing,'' Professor Craven said.
''The reality is that governors-general are at the end of their careers. Their reputations are absolutely vital. I don't think any reasonable person could perceive bias.''
Professor David Flint said there was no obvious conflict of interest and barrister Greg Barns said it was ''important that she shows that the office is above influence from any quarter'' by exercising her powers.
But ethicist Leslie Cannold and Melbourne barrister Peter Faris, QC, have said Ms Bryce's family link is a clear case of perceived bias.
Mr Faris suggested she hand her role over to the Chief Justice of the High Court.
However, this might be a controversial option - former governor-general Sir John Kerr approached High Court Chief Justice Sir Garfield Barwick before he dismissed prime minister Gough Whitlam in 1975.
Present Chief Justice Robert French has said that neither he nor other High Court judges would advise the Governor-General in case the matter ended up before them in court.
The meeting of Sir John and Sir Garfield on November 10, 1975, ''was, and remains, a controversial matter but, if only on that account, will not happen again'', Justice French said, quoting another of his predecessors, Sir Gerard Brennan.
He suggested in October last year that ''a small group of independent experts, perhaps even including one or more retired justices of the High Court, could be established for the purpose''.
In Tasmania's recent constitutional crisis, Governor Peter Underwood asked incumbent Premier David Bartlett to form government despite Mr Bartlett's public statements that the opposition should do so.