Britain’s hopes of avoiding a damaging no-deal Brexit hinge on talks between the Prime Minister and Opposition Leader after the House of Commons rejected a plan to vote on alternate plans.

John Bercow, Speaker of the House of Commons, used his casting vote to reject a third round of so-called indicative votes on an alternative Brexit deal after MPs tied for the first time in 26 years.

MPs voted 310-310 on the issue before Mr Bercow voted against it in accordance with the conventions of the House.

It came as Prime Minister Theresa May met opposition leader Jeremy Corbyn met on Thursday morning (Australian time) and agreed a “programme of work” to try to find a way forward.

The opposing Conservative and Labour parties have each appointed a negotiating team in a bold new strategy that aims to breach the Brexit stalemate.

A spokesman for Ms May said both sides were “showing flexibility”.

Mr Corbyn, however said the initial meeting was “useful, but inconclusive”, adding there had not been “as much change as [he] had expected” in Ms May’s position.

Mr Corbyn said he raised a number of issues with Ms May, including future customs arrangements, trade agreements and the option of giving the public another referendum.

Talks between the two parties will continue throughout Thursday.

Britain was supposed to leave the EU last Friday, but nearly three years after Britons narrowly voted for Brexit in a referendum, it is still unclear how, when or even whether it will quit the bloc.

After her EU withdrawal deal was rejected three times by MPs, with parliament and her Conservative Party hopelessly divided over Brexit, May said she would talk to the Labour Party leader in a bid to overcome what is now a national crisis.

“There are actually a number of areas we agree on in relation to Brexit … what we want to do now is to find a way forward that can command the support of this House and deliver on Brexit,” Ms May told parliament.

However, by approaching Mr Corbyn, a veteran socialist loathed by many Conservatives and mocked by Ms May herself as unfit to govern, she risks further inflaming divisions in her party. Two junior ministers quit on Wednesday.

“It now seems that you and your cabinet have decided that a deal – cooked up with a Marxist who has never once in his political life put British interests first – is better than ‘no- deal’,” Nigel Adams said as he resigned as a minister for Wales.

Ms May turned to Labour after a hardcore eurosceptic group of Conservatives repeatedly rejected her divorce deal, saying it would leave Britain a “vassal state”.

Labour wants to stay in a customs union with the EU, raising the likelihood of a “soft” Brexit option that keeps Britain’s economy closely aligned to the world’s biggest trading bloc.

Brexit Secretary Stephen Barclay said the government would accept a soft Brexit if parliament voted for it.

Ms May said on Tuesday she would seek “as short as possible” a delay to the current Brexit date of April 12, having repeatedly said she did not want Britain to have to take part in European Parliament elections on May 23.

Irish Prime Minister Leo Varadkar said he believed EU leaders were open to further delay and German Chancellor Angela Merkel said she would “fight until the last minute” for an orderly British exit.

But European Commission president Jean-Claude Juncker said Britain would not get any further short delays unless its parliament ratified a deal by April 12 – the date set by EU leaders as the effective cut-off for avoiding the European Parliament elections.

Meanwhile Bank of England governor Mark Carney warned on Wednesday that Britain still faced an “alarmingly high” risk of a no-deal Brexit, which could happen suddenly and by accident without politicians fully intending it.

-with AAP