However, the 27 insisted in their guidelines on the future trade deal that negotiations could only progress once the United Kingdom provides a legally watertight alternative to its "backstop" solution of keeping Northern Ireland in regulatory alignment with the EU.
The UK is due to leave in March 2019 and negotiators have said they want a deal in place by the end of the year. "I believe this is a new dynamic now in the negotiations", said Theresa May.
The guidelines, which were leaked to Business Insider earlier this week, set out the political terms of the withdrawal agreement and call for movement from Britain on the Northern Ireland border and the future of Gibraltar.
"We have already started bilateral talks with the United Kingdom", Rajoy told reporters in Brussels after European Union leaders approved guidelines for the next phase of Brexit negotiations.
At a summit in Brussels, 27 leaders without Britain backed the red lines for talks on the future relationship including trade, and approved a deal for a 21-month transition period.
"I believe we are approaching this with a spirit of co-operation, a spirit of opportunity for the future as well, and we will now be sitting down and determining those workable solutions for Northern Ireland, but also for our future security partnership and economic partnership", she said.
As well as future trade deals, the negotiating guidelines lay out how future security, defence and foreign policy ties will work.
In exchange, the Britain will be allowed to negotiate new global trade deals during the period at the same time as retaining the benefits of the single market and customs union.
But the text carried a warning that "nothing is agreed until everything is agreed" - a threat to May to avoid "backsliding" on a deal to let Northern Ireland remain regulated by Brussels if no better way is found to prevent a "hard border" that could risk the peace.
The Institute of Directors' Allie Renison said: "With the clock ticking and a fairly short implementation period, it is essential that we move into substantive trade negotiations as quickly as possible".
Leo Varadkar, Ireland's prime minister, welcomed Mrs May's acceptance that a backstop plan for the border would be included in Britain's withdrawal agreement.
But she said what was needed was a permanent exemption, given the "huge importance" of the industry to the British economy.
The Brexit schedule assumes the bloc and London would agree on the divorce deal, the transition and a framework for future trade in time for the 27 European Union leaders to endorse it at a summit in October, and have it ratified before Brexit on March 29, 2019.