"We have agreed a programme of work to ensure we deliver for the British people, protecting jobs and security".
Cooper said the legislation was still needed to set out a clear process for how decisions are taken over the length of any extension.
"This debate, this division, can not drag on much longer", May said in a televised statement from 10 Downing St. after an all-day Cabinet meeting.
Almost three years since the United Kingdom voted to leave the EU in a shock referendum result, British politics is in crisis and it is unclear how, when or if it will ever leave the European club it first joined in 1973.
The EU's chief Brexit negotiator Michel Barnier on Thursday welcomed the cross-party effort to resolved the deadlock.
"When you enter into a negotiation like this to find a compromise way forward, both parties have to give something up".
"Barnier stated" we could still aspire to prevent it" when London made a breakthrough before the April 10 European Union summit.
A Brexit-related vote in Britain's House of Commons on Wednesday ended in a tie, the first time that has happened in a quarter-century.
But May has failed on three occasions to pass her EU Withdrawal Agreement, forcing a delay to Brexit until at least 12 April and leading to some for call for the whole divorce to be cancelled altogether.
The plan will dismay the hardline Brexiteers in her own party, who for so long have opposed her Withdrawal Agreement - and may now, as a result, lose the hard Brexit they crave.
They said that the WA itself (on citizens' rights, money and the Northern Ireland border) was not renegotiable but there was leeway to change the political declaration (PD) on the future relationship, for example to choose a softer kind of Brexit with a closer UK/EU relationship.
"If we look like reaching any other decision than confirmatory vote that would be in breach of the decision made unanimously by conference in Liverpool and overwhelmingly supported by our members and it needs to be put to a vote by the shadow cabinet", the letter said.
In either event, May said she would ask the EU for a further short extension to Brexit in the hope of getting an agreement passed by Parliament before May 22, so that the United Kingdom does not have to take part in European elections. She also came under attack from a number of her own Conservative MPs for agreeing talks with the Labour leader.
May has always insisted Britain must leave the EU's single market and customs union in order to forge new trade deals around the world.
Pro-Brexit Conservatives, meanwhile, are angry at the prospect of a softer Brexit that keeps Britain bound closely to European Union trade rules and standards.
"We're in a situation where the expressed will of parliament is for some form of deal, so to put it in the double negative: parliament is against no deal", he said.
Firms have cautioned that the financial effect in Britain might be devastating.
If things remain unchanged, a no-deal Brexit in which tariffs will be suddenly reinstated will become a reality.
There are so many other obstacles - including persuading voters that it makes sense to start preparations for European Union parliamentary elections in the hope of cancelling them at the last minute - that I "can find no MP outside of the payroll" who thinks May's plan will work. "Both sides must play ball".
Speaking to the European parliament on Wednesday, President of the European Commission Jean-Claude Juncker dismissed May's request for an extension of Article 50 until May 22.
Irish Prime Minister Leo Varadkar plans to meet in Dublin Thursday with German Chancellor Angela Merkel to discuss the Brexit impasse.