There is no sense Britain is under pressure to leave the EU quickly, Theresa May's spokeswoman said after the prime minister met with Donald Tusk.
The first meeting between Mrs May and the European Council president since she became prime minister was described as "friendly", with both agreeing they should work together to ensure a "smooth" Brexit process.
Mrs May felt the EU understood her need to take time to form a negotiating stance before triggering Article 50, the formal divorce procedure, her spokeswoman said.
"The main points that the prime minister made were about working together so that there was a smooth process for the UK leaving the European Union, that is why we are taking time to prepare for the negotiations," the spokeswoman told reporters.
Mrs May also told Mr Tusk Britain would be a "strong player" while it remained in the EU and would continue to stand firm on sanctions against Russia over its action in neighbouring Ukraine.
European Council president Donald Tusk has urged Theresa May to start the Brexit process "as soon as possible", saying: "the ball is now in your court".
The leaders of the other 27 EU nations will hold talks in Bratislava next week and Mr Tusk said they would "discuss the political consequences of Brexit" for Europe.
But as they met in Downing Street, he told Mrs May: "It doesn't mean that we are going to discuss our future relations with the UK in Bratislava, because for this - and especially for the start of the negotiations - we need the formal notification, I mean triggering Article 50.
"I'm aware that it is not easy but I still hope you will be ready to start the process as soon as possible."
The prime minister said she wanted a "smooth" Brexit process and told Mr Tusk they had "serious issues" to discuss.
Poland's ruling Law and Justice party leader Jaroslaw Kaczynski has blamed European Council President Donald Tusk for Britain's vote to leave the EU, and called for him to disappear from European politics.
Mr Kaczynski told local media: "A particularly dark role was played by Donald Tusk, who conducted negotiations with the British and in fact contributed to them getting nothing.
"Hence, he is directly responsible for Brexit and should simply disappear from European politics. But this concerns the whole of European Commission in its present composition."
Donald Tusk, the president of the European Council, said Brexit was an "historic moment" but "not a moment for hysterical reactions".
He said the EU was prepared for this "dramatic scenario".
Mr Tusk added: "We are determined to keep our unity as 27."
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The incoming president of the European Council, Donald Tusk, has pledged to address the concerns of Britain in his new role.
He said the EU and he personally would take on the concerns voiced by Britain about the EU. "No reasonable person can imagine the EU without the UK," he said.
Prime Minister David Cameron has promised a referendum on continued EU membership by 2017.
Poland's prime minister Donald Tusk has been chosen by European Union leaders to become the new president of the European Council.
The top official will chair EU summits and mediate between the 28 governments for the next two-and-a-half years.
Federica Mogherini, the Italian foreign minister, has also replaced Catherine Ashton as the EU's foreign policy chief.
The Polish prime minister has told David Cameron that governments "cannot stigmatise any national group and must respect European law".
Donald Tusk's office described a phone conversation between the pair this afternoon as "an open and frank exchange of views".
It said Mr Cameron explained he had not intended to stigmatise Polish people who work in the UK when he made comments about child benefits payments to migrants.
But Mr Tusk was said to have told the Prime Minister that Poland could not accept such statements, regardless of their original intentions.