Embattled British Prime Minister Theresa May will face a vote of no-confidence triggered by her own party lawmakers over her controversial Brexit deal, as she vowed to fight it out with "everything I have got".

"I will contest that vote with everything I have got," May said in a statement at Downing Street, warning that the leadership challenge will delay or even cancel Brexit.

"I stand ready to finish the job," she said.

The vote, triggered amid ongoing divisions over her Brexit deal, takes place in the form of a secret ballot on Wednesday evening. May will need to convince a majority of MPs (158) to be able to win the vote and then her leadership cannot be challenged for a year.

However, if she loses, the party will have to elect a new leader who will then go on to become the next British Prime Minister.

The leadership challenge comes as she was desperately trying to rescue her Brexit deal as she tried to convince European Union (EU) leaders to offer some concessions to convince Britain's MPs to vote for it on Tuesday, a day after she postponed a crucial parliamentary vote scheduled for this week over the Withdrawal Agreement struck with the EU.

She was due to travel to Dublin on Wednesday but will now remain in London to contest the no-confidence vote.

European Commission President Jean-Claude Juncker, who held talks with the British PM in Brussels on Tuesday, said the EU would not "renegotiate" the deal but there was room for "further clarifications".

"The deal that we have achieved is the best deal possible, it is the only deal possible," he reiterated.

Britain's MPs have to give the go-ahead for May's deal if it is to come into effect when the UK leaves the EU on Brexit Day March 29, 2019.

But deep divisions remain on all sides of the House of Commons over the so-called "backstop", a temporary customs arrangement designed to prevent the need for checkpoints at the Irish border if a long-term solution between the UK and EU cannot be agreed post-Brexit.

Critics of the arrangement are unhappy that under the terms of the legally binding Withdrawal Agreement, the UK cannot exit the backstop without the EU agreeing to it, which could effectively leave the UK bound by EU customs laws beyond Brexit.

Downing Street had said a House of Commons vote on the agreement will be held on the deal before January 21, as that is the deadline enshrined in the European Union (Withdrawal) Act 2018. That deadline now looks more and more uncertain.