British Prime Minister Theresa May's attorney general was holding talks in Brussels on Tuesday in a last-ditch bid to secure changes to get her Brexit deal through Parliament.
Britain is due to leave the European Union in 24 days, but Parliament's rejection of Mrs May's deal in January has put in doubt how the UK's biggest foreign and trade policy shift in more than 40 years will take place.
She has told Mr Cox and Brexit Minister Stephen Barclay to secure changes to the Irish backstop, an insurance policy to prevent a "hard border" between Northern Ireland and Ireland if a future trading relationship falls short.
Mr Cox and Mr Barclay were meeting EU's chief negotiator Michel Barnier and other officials from the bloc, followed by further discussions to press what Mr Cox called "very constructive dialogue".
Time is of the essence, with some businesses increasingly concerned over the risk of a disorderly Brexit, which German car manufacturer BMW said on Tuesday could mean it would move some production of engines and its Mini model out of Britain.
Earlier, Foreign Minister Jeremy Hunt said the government still wanted "to leave at the end of this month and it depends how quickly we can get a deal through".
"Our ask of the EU is an important ask," Mr Hunt said. "We need substantive changes that will allow the attorney general to change his advice to the government that says that at the moment, theoretically, we could be trapped in the backstop indefinitely."
Mrs May has struggled to convince the EU that she can get the deal through a deeply divided Parliament in London, where MPs are increasingly trying to influence Britain's departure from the bloc.
She has offered legislators the chance to seek to prevent a no-deal departure and to delay Brexit if Parliament rejects the deal in a vote she has promised to hold by March 12.
British and EU officials have said any delay would probably be just for a few months.
Ministers are hoping Mr Cox can convince sceptic MPs that he has done enough to remove the threat of Britain ending up in the EU's Customs union indefinitely, something Brexit supporters say would make a mockery of the 2016 vote to leave the bloc.
Asked after a meeting of senior ministers whether Mrs May had set out specific details of what Mr Cox was seeking, her spokesman said: "No. You can expect them to be having detailed discussions around the legally binding changes we're seeking to the backstop, but it wasn't discussed at Cabinet."
Mr Cox, on arriving in Brussels, also declined to comment on details of the "sensitive discussions".
