Government sources have said the prime minister is set to tell MPs about the delay in a statement at 15:30 GMT.
Downing Street had been insisting the vote would go ahead.
The pound fell sharply in response, shedding 0.5% versus the US dollar to stand at $1.26, an 18-month low. The pound was 0.8% down against the euro.
Mrs May’s Commons statement will be followed by a statement from Commons leader Andrea Leadsom – and then a statement from the Brexit secretary Stephen Barclay on Article 50 – the legal mechanism taking the UK out of the EU on 29 March.
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However, amid uncertainty about the official procedure for calling off the vote, the BBC’s Laura Kuenssberg said it was not impossible Labour could join forces with Tory Brexiteers to try and make it go ahead.
DUP deputy leader Nigel Dodds said the situation was “quite frankly a bit of a shambles” and the PM was paying the price for crossing her “red lines” when it came to Northern Ireland.
Theresa May’s deal has been agreed with the EU – but it needs to be backed by the UK Parliament if it is to become law ahead of the UK’s departure.
Mrs May is thought to be trying to convince MPs to back her deal by suggesting the Northern Ireland backstop – the main item they object to – could be modified.
Mrs May has also been speaking to EU leaders about re-opening the withdrawal agreement, something both sides have previously ruled out.
It comes as the European Court of Justice ruled the UK could cancel Brexit without the permission of the other 27 EU members.
But European Commission spokeswoman Mina Andreeva said the EU would not renegotiate the Withdrawal Agreement.
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In a press briefing, she said: “We have an agreement on the table which was endorsed by the European Council in its Article 50 format on 25 November.
“As President Juncker said, this deal is the best and only deal possible. We will not renegotiate – our position has therefore not changed and as far as we are concerned the United Kingdom is leaving the European Union on 29 March, 2019.”
Dozens of Conservative MPs had been planning to join forces with Labour, the SNP, the Lib Dems, Plaid Cymru and the DUP to vote down Mrs May’s deal.
The Tory rebels and the DUP do not like the Northern Ireland “backstop”, a legally-binding proposal for a customs arrangement with the EU, which would come into force if the two sides’ cannot agree a future relationship which avoids the return of a visible Northern Ireland border.
Tory MPs say it is unacceptable because it would result in new regulatory barriers between Northern Ireland and the rest of the UK and could continue indefinitely, because the UK would not be able to leave without the EU’s approval.
Labour leader Jeremy Corbyn, who was hoping to use a defeat for Mrs May on Tuesday to force a general election, said: “The government has decided Theresa May’s Brexit deal is so disastrous that it has taken the desperate step of delaying its own vote at the eleventh hour.”
He said the prime minister should have “have gone back to Brussels to renegotiate or called an election” when it became clear she would not get her deal through Parliament.UP leader Arlene Foster said she had told the prime minister in a phone call that