Liberal MPs say the next Speaker must be able to restore respect for the role and the parliament as a whole.
The coalition party room will meet in Canberra on Monday to decide on who will be put forward for Speaker to replace Bronwyn Bishop.
The House of Representatives will then elect a Speaker as its first order of business.
Betting agencies have four Liberal MPs at short odds: Tony Smith, Russell Broadbent, Andrew Southcott and Philip Ruddock.
Queensland MPs Teresa Gambaro and Jane Prentice have also been named as potential candidates, but Ms Gambaro has told AAP she will not be running.
Prime Minister Tony Abbott says he spoke with Dr Southcott while in Adelaide for a cabinet meeting, but would "try and avoid a so-called captain's pick".
Ms Gambaro declined to say whether she would put up her hand, but said she believed the "best person would be selected for the role".
A senior Liberal figure said it was crucial that whoever was selected was a moderate who could restore respect for the role of Speaker and faith in parliament.
The source told AAP it was important the public got engaged in the discussion and not left it to a few Liberal powerbrokers.
Emily's List convenor Tanja Kovac said that as the Speaker saga was playing out there was another conversation among Liberal women about the poor support they were being given within the party.
"The challenge for women in the coalition is they need to step up if they are ready to assume leadership roles," she told AAP.
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