On Monday, the bullying and harassment of House of Commons staff report led by Dame Laura Cox was published.
Citing well-placed sources, the BBC reports Mr Bercow has told friends and close associates about his plan but has not decided how and when to officially announce his departure.
The chairwoman of the Women and Equalities Committee, Maria Miller MP, said the report showed "bullying and harassment is coming right from the top" and it is not right for Mr Bercow to oversee reform.
Approached for a comment on the Speaker's plans, a spokeswoman said: "The Speaker was elected by the House in 2017 for the course of the Parliament".
In a short statement at the start, Mr Bercow said it was clear that decisive action was needed and Dame Laura's main recommendations accepted.
Mr Bercow himself has been accused of bullying and intimidating two members staff, dating back to 2009, and the report slams the current system for addressing allegations as MPs often make decisions about cases which may involve themselves. "We have outstanding allegations directly against the Speaker".
Mr Bercow has faced calls to resign in the wake of the report, which found "urgent and serious problems" in the way abusive behaviour by MPs and staff is dealt with.
Writing in The Times, he said: "The change in culture has to come from the top, and unfortunately I no longer believe that the Speaker, John Bercow, is the correct person to provide that leadership, so he should step down".
It also includes shock claims of "inappropriate touching" happening to women in parliament, including male staff "trying to kiss them, grabbing their arms or bottoms or stroking their breasts or bottoms".
Green MP Caroline Lucas called for "independent processes, not innuendos" while Labour's Jess Phillips said she, unlike other MPs, had actually spoken to those affected by bullying.
Commons leader Andrea Leadsom said the new complaints procedure was working well but it was "far from job done" and harassment must be "stamped out" completely.
However, others warned against a witch-hunt, with ex-Conservative minister Sir Desmond Swayne saying most MPs acted "perfectly properly" and were not treated like "demi-gods" by their staff.