According to PoliticsHome, Dame Rosie announced her bid this morning.
In an email, seen by the website, Dame Rosie, who has been a deputy Speaker since 2017, officially announced she was in the running to replace Mr Bercow who is standing down on October 31.
The Doncaster Central MP said: "Parliament has been through a turbulent period and the role of Speaker has been under enormous scrutiny.
"Speaker Bercow’s period in office has been historic in many ways, and not without controversy. I believe I can be a stabilising, unifying Speaker who would be able to balance the rights of the Executive to carry out its mandate where it is given by the people, with the rights of parliamentarians to hold the government to account, scrutinise legislation and to debate issues of the day.
"Getting that balance right is vital to our democracy."
"It would be a huge privilege to be that Speaker and I ask for your support to do so."
Ms Winterton becomes the tenth MP to announce they are running to be Speaker.