British Prime Minister Theresa May's government plans to introduce a bill in parliament implementing the Brexit deal in the first week of June, a spokesman confirmed late Tuesday.
"This evening the prime minister met the leader of the opposition in the House of Commons to make clear our determination to bring the talks to a conclusion and deliver on the referendum result to leave the EU," the spokesman said.
"We will therefore be bringing forward the Withdrawal Agreement Bill in the week beginning the 3rd June. It is imperative we do so then if the UK is to leave the EU before the summer Parliamentary recess.
"Talks this evening between the prime minister and the leader of the opposition were both useful and constructive," the spokesman added. The discussions were to resume Wednesday at an official level.
May's Brexit deal with the EU has been rejected by the British parliament three times.
London's original Brexit date of March 29 came and went without British parliamentary approval.
Should British MPs approve the Withdrawal Agreement Bill that May seeks to introduce in June – needed to make the Brexit deal British law – parliament would not have to vote again on her deal with the EU.
The British prime minister chaired a three-hour Cabinet meeting Tuesday to discuss progress on resolving the political impasse. Downing Street said ministers "agreed to continue discussions with Labour to see what was possible."
It said the Cabinet discussed possible compromises by the government to secure the Labour leadership's backing for a deal.
The ministers agreed that it was vital to steer the government's EU Withdrawal Agreement Bill though parliament before the summer recess.
But Labour's Shadow Chancellor John McDonnell said he had seen no "significant shift" by the government during the talks.
"We're not near what we want," McDonnell, a close ally of Labour's left-wing leader Jeremy Corbyn, told a Wall Street Journal conference in London.
Fourteen influential Eurosceptic Conservative lawmakers warned May earlier Tuesday not to agree a deal with Labour.
"A customs union with the EU is both bad policy and bad politics," the group wrote in an open letter to May published by The Times.
Led by Graham Brady, chairman of the 1922 Committee of Conservative lawmakers, they warned that a deal with Labour would "split the party."
Brexiteers have warned May that such an agreement would lose "the loyal middle of the Conservative Party, split (the) party and with likely nothing positive to show for it."
"No leader can (bind) his or her successor, so the deal would likely be at best temporary, at worst illusory," they added.
The signatories included former Foreign Secretary Boris Johnson and former Brexit Secretaries David Davis and Dominic Raab, all of whom resigned to oppose May's Brexit plans.
Johnson and Raab are among the front-runners to succeed May.
"I signed this letter because the customs union option robs us of the opportunities of Brexit, and breaches the cardinal promise made in our 2017 election manifesto," Raab tweeted later Tuesday, adding that he will "vote against (a customs union deal) every time."