This Week presenter Neil implied the speaker was himself courting publicity and passing judgement that was "above" him.
Mr Bercow caused a political storm after he said he would try to ban Donald Trump from entering Parliament and accused the Republican of "racism and sexism". The former Tory MP may now face a vote of no confidence as Speaker as up to 150 MPs backed the motion.
The former Tory MP may now face a vote of no confidence as Speaker with reports suggesting as many as 150 MPs could back the motion.
Taking a dig at Mr Bercow as he began his weekly late-night show, Neil jokingly suggested he wouldn't allow Mr Trump onto This Week.
Andrew Neil mocks John Bercow's comments on President Trump
I would not wish to issue an invitation for President Trump to appear on this programme
He said: "Let me make clear right from the start that though we value our relationship with the United States and whether or not Donald Trump makes his State Visit to this country is way above my pay grade.
"I would not wish to issue an invitation for President Trump to appear on this programme. I appreciate he's expressed no interest in doing so anyway, but that won't stop me doing a bit of grandstanding and virtue signalling before the rest of the media and the wider public."
He went on to mockingly add that despite welcoming Vlad the Impaler, Diane Abbott, Adolf Hitler, Attila the Hun, Ken Livingstone and Count Dracula to his studio over the years, an invite would not be extended to non-drinker Donald Trump.
Mr Bercow's position as Commons Speaker is under scrutiny 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.
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The Conservatives must realise that being sceptical is different from being phobic in what is an interdependent world.
Referring to the Speaker and then United States President Trump, Neil quipped: "Let me say this as a self-important, solipsistic, egotistical, preening, pretentious, puffed up, postulating, pompous little pixie in love with the sound of my own voice.
"This programme'slong-runningg opposition to free speech, sensible debate and proper analysis, coupled with his deep commitment to free Blue Nun for all, means I would not feel comfortable sharing a sofa on the public airwaves with someone who says what he thinks and has never even had a sip of alcohol in his life."
Mr Bercow has previously been accused of using a controversial loophole in the Freedom of Information Act to avoid releasing information about the level of drinking in Parliament. MPs have in the past bought thousands of bottles of the Speaker's whisky.
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A vote of no confidence motion has been put forward by Tory backbencher James Duddridge, who responded furiously to Mr Bercow's comments on Monday.
Mr Duddridge said the Speaker's comments were just the latest in which he ignored his position's expectation of political neutrality.
Although Mr Bercow won applause from some MPs, Mr Bercow has apologised to his House of Lords counterpart, Lord Fowler, for not consulting him before making them.
Mr Trump's state visit could take place around June.
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