LONDON, Jan.30 (Xinhua) -- British Prime Minister Theresa May and the main opposition Labour leader Jeremy Corbyn, held their first face-to-face talks Wednesday to discuss Brexit.

Hours after the two front bench politicians had clashed in the House of Commons over Brexit during Prime Minister's Question, they held private talks at Westminster.

Media reports in London later said Corbyn told May it was not acceptable for May to keep a no-deal option on the table.

Corbyn had shunned talks with May earlier because she refused to rule out Britain leaving the European Union with no deal.

A spokesman for Corbyn said the two leaders had a serious exchange of views, exploring a customs union and a single market relationship, describing it as a very cordial meeting. Both have agreed to further meetings.

May was buoyed by a majority vote in the House of Commons on Tuesday night, backing her Brexit deal if the so-called Irish border stopgap measure is removed.

It puts her Brexit deal within grasp, if she can persuade EU leaders to agree to change their insistence on measures to prevent a hard border between Northern Ireland and the Republic of Ireland.

The signs coming from Brussels Wednesday did not give May any comfort that she was close to finally clinching a Brexit deal.

European Commission President Jean-Claude Juncker warned in Brussels Wednesday that a no deal Brexit was more likely after British MPs voted to send May back to Brussels to renegotiate the Irish border backstop.

Juncker and the EU's chief Brexit negotiator Michel Barnier both said in the European Parliament in Brussels that they would reject any British efforts to renegotiate the Irish border backstop.

Juncker told the European Parliament that Brexit was a bad decision, but he ruled out renegotiating the Irish backstop.

He said: "The withdrawal agreement remains the best and only deal possible. The debate and votes in the House of Commons yesterday do not change that. The withdrawal agreement will not be renegotiated."

Seasoned political observers say it is now a case of who blinks first, London or Brussels, as the waiting game continues.

Nigel Dodds, leader of the 10 Democratic Unionist Party (DUP) politicians at Westminster, used Prime Minister's Questions in the House of Commons to accuse the Irish Government of highly reckless talk over the potential of a no-deal Brexit to cause trouble at the border between Northern Ireland and the Republic of Ireland.

He said: "In recent days we have heard the Irish Prime Minister talk about bringing his troops up to the border in the event of no deal. We have heard the Irish Deputy Foreign Minister talking of people jumping out of windows. Is not this highly reckless talk extremely dangerous in the present circumstances?

"That sort of rhetoric should be toned down and we should instead focus on what Michel Barnier said the other day, that even in the event of no deal, we would sit down and find operational ways to have checks and controls away from the border. Is not that the way forward? And it blows a hole in the entire concept of this backstop."

May told Dodds: "We have obviously sent a clear message about what needs to happen in relation to the backstop. We retain our commitment to no hard border between Northern Ireland and Ireland, and look to working with the Government of Ireland and with the European Union to ensure that we can all maintain our commitments under the Belfast/Good Friday agreement and that commitment to no hard border on the island of Ireland."

May plans a series of meetings started Thursday with politicians from her Conservative Party to consolidate the support she received Tuesday night.

May now has several weeks to persuade the EU to change its mind on the Irish backstop arrangement. If there is movement by the EU, May then hopes MPs will give their final agreement to a Brexit deal.

She is scheduled to present her new Brexit plan to the House of Commons mid-February. If MPs and the EU are in agreement the way will be clear for Britain to leave the bloc on March 29.

If May's Brexit plan collapses, it opens up a box of alternatives, including Britain leaving on March 29 with no deal, a snap election or another confidence vote.