Ahead of the controversial vote, Mr Bercow faced a backlash from angry Conservative MPs after he selected the amendment from former Tory minister Dominic Grieve, to ensure the PM returned with a revised European Union exit plan within three sitting days should her original Brexit deal be defeated.
While the two votes are largely symbolic, the successive defeat shows how weak parliamentary support is for Mrs May's Brexit plans.
The Pro-Remain warning comes amid widespread expectations that the Government is heading for defeat in next week's "meaningful vote" on the deal, with opposition from both pro-Leave and pro-Remain Tories. "The idea is to hope for the best and prepare for the worst", she added.
She has promised to secure further assurances from the EU on the most controversial elements of the Brexit deal relating to Northern Ireland, and has held talks with European leaders in recent days.
There were turbulent scenes in parliament when some in May's Conservative Party accused the speaker of bias.
Today's debate will be opened by the Brexit Secretary Stephen Barclay and closed by the Prime Minister next Tuesday.
But her spokesman added: "If that were not to take place. we would respond quickly and provide certainty on the way forward".
It was the second setback in as many days for the prime minister, after MPs voted late Tuesday to deny the government certain taxation powers in a no-deal scenario - another attempt to avoid such an outcome.
The government lost by 11 votes, with 297 MPs siding with them and 308 against.
Handing the Stormont Assembly a "veto" on Northern Ireland's post-Brexit trade rules would be unacceptable, Sinn Fein has said. There's no guarantee Labour could marshal the majority support in Parliament needed to topple the government - and even if it did, a new election would not automatically stop the Brexit countdown clock.
The Brexit Withdrawal Agreement covers the terms of the UK's exit and the framework of its future relations with the EU.
He said: "There now is a withdrawal deal on the table and it hasn't been well received". Sammy Wilson, the party's Brexit spokesman, said: "The only thing which could swing the DUP round is if the backstop as it applies to the United Kingdom as a whole or to Northern Ireland specifically were removed from this agreement".
But the DUP, which is propping up Mrs May's minority government through a confidence and supply agreement, immediately rejected the offer.
The Dutch move is aimed at calming fears of chaos for people and businesses in the event that there is no withdrawal agreement for Britain to leave the bloc on 29 March.
The vote, which saw 20 legislators from May's Conservative Party rebel and side with the opposition, indicates that a majority in Parliament opposes leaving the European Union without an agreement and will try to stop it happening.
Opposition to May's deal is spread across the political spectrum, including sections of her own ruling Conservative Party and the 10 Northern Irish Democratic Unionist Party (DUP) MPs her government relies on to command a majority in parliament.
"I don't think that the British public are served by fantasies about magical alternative deals that are somehow going to sort of spring out of a cupboard in Brussels", Lidington told the BBC.