The Justice Secretary was quizzed about the issue following a speech but declined to speak in support of Mr Bercow, who told students at Reading University that he voted Remain.
Tory MP James Duddridge has since tabled a motion of no confidence in the Speaker, who is traditionally meant to remain impartial.
Campaigners have reportedly claimed more than a dozen cabinet ministers will back the motion.
When asked if she was one of those ministers, Ms Truss replied: “In terms of the Speaker, that is a matter for Parliament, as the leader of the House made very clear yesterday.”
Liz Truss said John Bercow's future was a "matter for Parliament"
n terms of the Speaker, that is a matter for Parliament, as the leader of the House made very clear yesterday
Theresa May also refused to back Mr Bercow after her spokesman insisted it was a “matter for MPs”.
“He’s the elected chairman of the House of Commons, he's not a member of the government, he's not appointed by the Government,” he said.
"He represents all the MPs in the House of Commons. It's up to them to decide whether they feel he's doing a good job or not."
Speaking about Mr Bercow telling students he “thought it was better to stay in the European Union than not”, Mr Duddridge said his comments were “wholly unacceptable”.
John Bercow's best quotes
Wed, February 8, 2017
Outspoken John Bercow, the Speaker of the Commons, has both wise and not so wise words.
Get Quotes on Home Insurance
1 of 9
The Conservatives must realise that being sceptical is different from being phobic in what is an interdependent world.
A motion of no confidence has been tabled against the Commons Speaker
He told BBC Radio 4’s Today programme on Monday: “He was miked up, he knew he was being recorded, he knew he was overstepping the mark and this is wholly unacceptable. We expect our speaker to be impartial.
“In the House of Commons, he quite frequently calls his friends rather than being totally impartial but critically now, he can no longer be impartial given he’s expressed views so publicly and so clearly.”
But the Speaker’s spokeswoman said he had voted in June's referendum "along with millions of others".
She said: ”The record shows that he has rigorously facilitated the raising of concerns of those on both sides of this argument, as he does on every other issue.
"The Speaker's impartiality is required on matters of debate before the House, and he has been scrupulous in ensuring that both sides of the argument are always heard."