Senior UK government ministers held secret talks with rival Labour members of parliament to secure bipartisan support for a soft Brexit, British broadsheet "The Daily Telegraph" revealed on Tuesday.

Senior figures in Prime Minister Theresa May's team discussed with their Labour counterparts how to force May to make concessions on immigration, the customs union and the single market, it reported.

They also discussed the formation of a cross-party Brexit Commission to agree common ground between the parties and ensure an orderly withdrawal from the European Union, the paper reported.

The MPs involved in the secret talks were mostly believed to be those who campaigned to remain in the EU, who had already forged alliances when they campaigned together in the lead-up to the referendum.

Michael Gove, former Brexit leave campaigner and now Environment Secretary, later confirmed to BBC television that he had discussed Brexit with Labour politicians.

"I talk to politicians from every party in order to make sure that we get the right approach," he told the show.

"Of course I talk to people from different parties, that's what governing in the national interest is all about."

Gove said he rejected the terms "soft" and "hard" Brexit because [he is] "never sure what they mean" and that the term "hard Brexit" was invented by people who want Brexit to be seen as "some sort of punishment."

On Monday, Labour Party politician and former shadow home secretary Yvette Cooper called for a cross-party commission on Brexit, in an article for "The Guardian" newspaper in an article for "The Guardian" newspaper.

"After the referendum last year, I called for the government to approach this in a cross-party way to get the best deal. Now it is more important than ever. There is neither strength nor stability in a narrow, bunkered one-party approach; you need to include people with different ideas to get the best deal and widest support," she wrote.

"So we should set up a small cross-party commission to conduct the negotiations, and have a clear and transparent process to build consensus behind the final deal. It should be accountable to parliament but avoid getting caught up in the inevitable hung parliament political rows."

A similar call was made by William Hague, the former Conservative leader, who outlined a plan in the "Daily Telegraph" as to how a cross-party commission would work, saying it would have to include business leaders, the first ministers of devolved governments and "the leaders of all the opposition parties - yes, even Corbyn."

Tuesday's "Daily Telegraph" report said May had been "aware" of the secret talks for days but that she had so far done nothing to stop them.

Theresa May has been in damage control since a disastrous snap election result lost the Conservative Party its majority in parliament. They have since been negotiating a potential coalition with the hardline Northern Irish Democratic Unionist Party (DUP), as May's "hard brexit" position slips into a precarious position.

The DUP are deeply euroskeptic, but they have balked at some of the practical implications of a hard Brexit - including a potential loss of a "frictionless border" with the Republic of Ireland. Leaders planned to discuss efforts to minimize the potential damage to Northern Ireland during talks on Tuesday.

A deal with the DUP could also risk destabilizing the political balance in Northern Ireland by increasing the influence of pro-British unionists who have struggled for years with Irish Catholic nationalists who want Northern Ireland to join a united Ireland.

The European Parliament's chief Brexit negotiator Brexit Guy Verhofstadt said the EU wanted clarity from London as soon as possible on whether it intended to stick to its stance towards negotiations or alter it.

"We await... the position of the United Kingdom," the former Belgian premier told a news conference on Tuesday.

"It's unclear if the UK government will stick to the line that they had announced in the letter of the 29th of March or if they will change it... taking into account the outcome of the election."

EU Brexit negotiator Michel Barnier warned May not to waste time in naming a date for talks, lamenting that it was already three months since May had formally triggered the two-year exit process.

"My preoccupation is that time is passing, it is passing quicker than anyone believes because the subjects we have to deal with are extraordinarily complex... I can't negotiate with myself," Barnier was quoted as saying by the "Financial Times."

Formal negotiations between Barnier and British Brexit minister David Davis had been due to start next week but that timetable was thrown into doubt by May's catastrophic election result.