The front-runner to become Parliament's senior official is "totally out of her depth" and her appointment would be a mistake, former Commons speaker Baroness Boothroyd has warned.
The peer, who has passed a dossier detailing her concerns to William Hague, said Carol Mills lacks the knowledge for the "hugely important constitutional role" of Commons clerk.
Ms Mills, head of the Department of Parliamentary Services (DPS) of the Australian Senate in Canberra, is believed to have been picked by a cross-party selection panel for the prestigious unelected role.
Downing Street is due to ask Buckingham Palace for the Queen's formal approval of the chosen candidate to replace Westminster stalwart Sir Robert Rogers, who departs at the end of the month.
But demands for a rethink by Speaker John Bercow were intensified by the publication of details of an email reportedly sent by the clerk of the Australian senate, Rosemary Laing, casting serious doubt about Ms Mills' suitability.
Ms Laing was said by the Mail on Sunday to have sent an email expressing "disbelief and dismay" among colleagues at the potential appointment of someone "without parliamentary knowledge and experience".
The job of DPS is "essentially the role of an administrator and bears no resemblance to the role of a professional parliamentary officer", she is said to have warned Westminster officials.
Lady Boothroyd told BBC Radio 4's World At One that she had passed details of her concerns to Commons Leader Mr Hague, whose predecessor, Andrew Lansley, was on the cross-party appointment panel chaired by Mr Bercow.
She said: "My real concern, my deep concern, is - I'm very sorry for this lady, Ms Mills - because she is totally out of her depth, she has no experience, she has no knowledge of the constitutional role of a parliamentary clerk.
"I don't wish that upon the House of Commons and this is what has got to be revisited."
The peer said the memo from Ms Laing showed that Ms Mills " is not knowledgeable enough, she hasn't had the experience to do the job of a clerk".
She added: " Not only does the clerk advise the Speaker - that's very important - but the clerk is there to advise the Government, to advise the Prime Minister, members of the Cabinet, Leaders of the Opposition and any individual Member of Parliament.
"It's a hugely important constitutional role."
She claimed that Mr Bercow " may well want to feel that he can make his own decisions and not refer to anybody or not take anybody's advice".
She added: "Nobody knows better than I do the difficulties of the job of Speaker - you can't please everybody.
"But at the end of the day, any Speaker has to say what is 'good for Parliament' - that is the decision that has to be made, and I think a mistake has been made on this one."
She said if the aim was to separate the clerk's role as a constitutional expert from the duties as chief executive of the House of Commons then a committee of MPs should have examined the proposal.
The Cabinet Secretary has reportedly been asked to intervene in the selection process, according to a report in the Guardian.
Critics have accused Mr Bercow - who has attempted to drive a significant modernisation of the Commons - of overlooking highly-qualified deputy clerk David Natzler for reasons of political correctness.
The Guardian reported that Mr Natzler's requests to see documents from the selection process had led Sir Robert to ask Cabinet Secretary Sir Jeremy Heywood to step in.
Mr Bercow's spokeswoman defended the selection process.
"The Speaker is absolutely confident that the interview panel rigorously scrutinised everybody who came up for interview and gave everybody a fair opportunity to present themselves," she said.
"They carefully questioned everybody. It was a thorough and very fair recruitment process."
An initial eight candidates were interviewed by the six-strong panel chaired by Mr Bercow and including senior MPs from all three main parties, with three invited back for a further grilling.
The panel consisted of Conservative Mr Lansley - who subsequently lost his job as Commons leader in the reshuffle - shadow leader Angela Eagle, Labour chairwoman of the Public Accounts Committee Margaret Hodge and Liberal Democrat John Thurso.
Parliamentary and Health Service Ombudsman Dame Julie Mellor was the external panel member.
Mr Bercow had insisted on opening up the recruitment process which previously involved suitable successors being nominated by outgoing clerks.
But Tory MP Jacob Rees-Mogg said that despite being a fan of the Speaker's reform efforts, he felt the job should have been given to Mr Natzler who was "deeply qualified and knows the constitution backwards".
"Unfortunately the candidate who has emerged does not know the Australian constitution let alone the British," he told BBC Radio 4's Westminster Hour.
"She used, apparently, to run the Sydney Opera House.
"Now this doesn't sound like the level of qualification needed to be the most senior constitutional adviser not just to the Speaker but to every individual member of the House of Commons."
Downing Street said the recruitment process was a matter for Mr Bercow and the House.
"The process is under way. It hasn't been formalised yet," a No 10 spokeswoman said.
Asked if Mr Cameron was happy with the way the process had been conducted, the spokeswoman said: "It's a matter for the Speaker and the House.
"There is a well-established process and it follows the same format by which the current Clerk of the Commons was appointed."