There has been speculation Mr Bercow will announce his retirement this summer although he has not spoken about his plans publicly, always insisting he would tell MPs first.
When he was first elected in 2009, Mr Bercow said he intended to serve no more than nine years. He was re-elected unopposed again in 2015 and 2017.
In recent months, he has become a pivotal but controversial figure in Brexit debates in the Commons, angering and delighting MPs in equal measure by his interpretation of parliamentary conventions.
Referring to the controversy surrounding the Speaker’s impartiality, Mr Bryant said he thought MPs had “all been a bit bloodied and bruised of late, and the next Speaker has got to be somebody who is going to tend to those wounds a bit more”.
The Labour MP for Rhondda said he would stand for the role – which is elected by MPs in a secret ballot.
He said he would “do everything in my power not to belittle or diminish or lecture MPs from the chair, but, insofar as it is possible, to respect every single person”.
Mr Bryant also said he would like to open up the Speaker’s residence in the Palace of Westminster to welcome international guests and MPs’ spouses.
“I think spouses of MPs get a rough time, so I’d quite like to have some events in Speaker’s House which are either exclusively or primarily for MPs and their spouses, or maybe just for spouses,” he said.
Under a Chris Bryant speakership, parliamentary staff could find themselves in charge for the day. “I’d like to have some kind of event for the staff who run the building on the 6th January, with MPs serving,” he said.