Some shadow cabinet ministers have suggested that the counter-claims of bullying centre on Corbyn's chief of staff, Karie Murphy, which was strongly denied by the leader's office.
Debbie Abrahams, the party's welfare spokeswoman, denied the allegations and said she had been victimised by the Labour leader's office.
Mrs Leadsom announced her support for an inquiry in response to an urgent question from Green Party co-leader Caroline Lucas.
"People working in parliament should be able to work without fear of intimidation or bullying and today I'm calling for every member of staff to be given the best possible protection", she said.
The speaker chaired Monday's debate himself, rather than handing over to one of his deputies, a decision that drew criticism from Conservative MP James Dudderidge.
Ms Lucas said there should be an assumption that all historic allegations of bullying and harassment of Commons staff - known as "clerks" - should be investigated. She said she would propose a short independent inquiry to the House of Commons commission, the body that runs the Commons administratively, when it meets on Monday.
"Following the Newsnight allegations and others, it is clear the respect policy may not be sufficient to protect house staff", Leadsom said.
Elsewhere, Labour MP Jess Phillips told the Observer Mr Bercow should "consider stepping back" from his job to allow an independent investigation to take place.
Conservative MP James Duddridge - a long-term critic of Mr Bercow - questioned whether it was "appropriate" for Mr Bercow to chair the proceedings given he had been named in the BBC report.
News that there is to be an investigation into Parliamentary bullying is likely to pile further pressure on Bercow, following allegations that his private secretary resigned in 2011 and suffered PTSD after working for him.
Mr Farrelly told the Commons that he had been "flayed by selective leaking" and said the Newsnight report had been "one-sided".