"If those three points are not clarified, then we can not move forward on the rest".
However, speaking in Paris, Mr Macron said that while he welcomed Mrs May's "willingness" to move forward, more progress was needed on the rights of European Union citizens in Britain and the border with Ireland as well as the so-called "divorce settlement".
His cautious welcome to the speech came just a day after he had tried to turn the heat up on Mrs May over Brexit by demanding Britain "settle its accounts" with the bloc.
But the past three rounds of talks have yielded little progress, so British Prime Minister Theresa May used a speech in Florence, Italy on September 22 to present her proposals for the future to accelerate Brexit negotiations.
Addressing EU citizens in Britain she said: "We want you to stay, we value you and we thank you for your contribution to national life", and added that she wanted them to be able to continue living their lives in the same way.
Mrs May insisted that Britain would honour its commitments made in the budget up to 2020, saying that none of the other countries would need to pay more or receive less... as a result of our decision to leave.
Barnier, a former French foreign minister and European Union commissioner, is to host his British counterpart David Davis in Brussels on Monday for a fourth round of negotiations, and he said he expected details from Britain at that meeting.
On the economic front, May suggested it would not be necessary to negotiate a new trade deal with the European Union "from scratch".
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France, Germany and other European Union countries are insisting that Britain reach a deal with the bloc on the terms of its divorce before moving on to negotiate its future trading relationship.
"We believe we should be as open-minded as possible on how we work together on security matters", May said.
'The speech shows a willingness to move forward, as time is of the essence.' Mr Barnier added: 'Prime Minister May's statements are a step forward but they must now be translated into a precise negotiating position of the United Kingdom government'. "I am even more concerned now", he tweeted Friday.
In addition to agreeing to pay for projects that the United Kingdom has already committed to through the EU, May requested that the country pay contributions to take part in specific policies and programmes within science, education, culture and security. She is expected to say that it will take a "profound sense of responsibility to make this change work smoothly and sensibly".
She said if "we open our minds to new thinking and new possibilities we can forge a brighter and better future for all our peoples".
Mrs. May also suggested a two-year "implementation period" would be needed after the U.K.'s planned withdrawal in 2019 for both sides to prepare for new trading relations.
As Barnier said in Rome, what they want is clarity and a swift resolution at the negotiating table.
He said: "Fifteen months after the European Union referendum, the government is still no clearer about what our long-term relationship with the European Union will look like. We are not complacent about the challenges ahead but we are optimistic about our bright future", the spokesperson added.