"He doesn't want it to look as if he's been forced out," one source told the BBC.
"But it may be it won't be enough, and some people will insist he goes much sooner."
Mr Bercow, who was elected to the role in 2009, has regularly spoken about his Jewish upbringing.
He told a group of World Jewish Congress (WJC) delegates visiting Parliament last September of the “pernicious and insidious” threat still lurking against Jews.
In a 2015 interview, Mr Bercow said: “My father was Jewish and I went to Finchley Reform Synagogue and had a barmitzvah, although I am secular. My mother is not Jewish, but she converted.”
He was also the first Speaker of the House to pay an official visit to Israel in 2017 as a guest of Speaker of the Knesset, Yuli-Yoel Edelstein.
Describing the visit as a “great privilege”, Mr Bercow stressed the need for relations between the UK and Israel “irrespective of which government is in power at the time –either in Britain or Israel.”
Mr Bercow had originally promised to retire after nine years in office, which would have seen him depart in the summer of 2017 but he had expressed a desire to see out the Brexit process.
Asked to comment on the Speaker's plans, his spokeswoman said: "The Speaker was elected by the House in 2017 for the course of the Parliament. In the event he has anything to say on his future plans, he will make an announcement to the House first."
