Both men said they had spent a great deal of time discussing the question of Northern Ireland where all parties seek to uphold the free border.
- June 21 - The Queen's Speech sets out details of extensive UK Government legislation required to put Brexit into effect, expected to include a Great Repeal Bill bringing European Union laws and regulations onto the British statute book, as well as bills on issues including immigration and customs. The price of such a deal remains unclear.
The loyalist DUP has said it will not accept any "special status" for Northern Ireland in the European Union after Brexit, which would eliminate one of the leading suggestions for a solution. The talks must be completed and endorsed by parliaments by the end of March 2019.
Foster later said it was "right and proper" for the DUP to support May's legislative program, which is due to be announced next week and will be the first formal test of her ability to govern.
But UK government officials insisted they will continue to push for talks on a comprehensive trade agreement to discussed alongside a deal on the withdrawal process. She is seeking an arrangement with the Democratic Unionist Party in Northern Ireland.
However, the speech was cancelled following the Grenfell Tower tragedy in West London.
"But we want to keep the door open for the British".
After facing criticism for not meeting survivors on Thursday, May visited a hospital on Friday to speak to some of the people who escaped the fire.
But he warned that "we need to get there via a slope, not via a cliff edge".
Britain gave official notification that it would leave the European Union at the end of March, meaning it will leave the bloc, which is its biggest trading partner, in March 2019.
Brexiteers accept there is likely to be some short-term economic pain but say Britain will thrive in the longer term if cut loose from what they see as a doomed experiment in German-dominated unity and excessive debt-funded welfare spending.
"I hope that today we can identify priorities and a timetable that would allow me to report to the European Council later this week that we had a constructive opening of negotiations", he said.
EU Chief Negotiator Michel Barnier, welcomed UK's counterpart David Davis, Secretary of State for Exiting the EU, as the British have responded positively to Barnier's call to open the negotiations at a political level, a week after preparatory "talks about talks" at a civil service level.