Former foreign secretaries Jack Straw and Dame Margaret Beckett have added their voices to a cross-party campaign for Ms Mills to face a confirmation hearing of MPs before the Commons clerk appointment is made.
Meanwhile, Ms Mills has broken her silence over an email from the Australian clerk of the senate, Rosemary Laing, which warned that she had no "parliamentary knowledge or experience" and should not be appointed.
Ms Mills, head of the Department of Parliamentary Services (DPS) in Canberra, is believed to have been recommended for the prestigious role following a recruitment process with a selection panel led by Speaker John Bercow.
She said she was disappointed to read the missive from Ms Laing. "It would not be appropriate for me to comment further in a personal capacity at this time," she said.
"As a senior parliamentary officer, I take seriously my responsibilities to promote and uphold the values and code of conduct articulated in the Commonwealth of Australia Parliamentary Service Act 1999."
Ms Laing apparently cited an incident in which the Australian Department of Parliamentary Services confirmed it had used CCTV cameras to retrace the movements of a DPS employee and show her pushing an envelope under the door of a senator's office late at night.
The episode has been referred to the senate privileges committee for inquiry.
Referring to the Westminster role, Ms Mills said: "I have declined to confirm or comment upon my candidacy to date, as the selection process has not concluded."
Concerns have been raised about Ms Mills's ability to fulfil the role as adviser on parliamentary procedure and constitutional matters, with former speaker Baroness Boothroyd previously warning she would be "totally out of her depth".