In a speech in Florence, Mrs May offered to continue paying Brussels for a two-year transition period after the UK leaves in 2019, during which time EU nationals would be able to settle in Britain.
President of France Mr Macron said that while he welcomed Mrs May’s “willingness” to move forward, more progress was required on the rights of EU citizens and the border with Ireland.
The President said: “Before we move forward, we wish to clarify the issue of the regulation of European citizens, the financial terms of the exit and the question of Ireland.
“If those three points are not clarified, then we cannot move forward on the rest.”
Her proposal means that British taxpayers will pay around £18 billion more to Brussels on top of the final membership payments due to end in 2019.
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Mrs May had initially hoped that her speech would enable talks to move forward between Brussels and the UK.
The EU’s chief negotiator Michel Barnier said that he welcomed the “constructive spirit” of her speech but said it need to be translated into a “precise negotiating position”.
Tory backbencher Bernard Jenkin rejected this, saying: “How can we put money on the table when we don’t know what the deal is?
“How can we discuss how the finer details of the Northern Ireland border should operate when the EU won’t even begin to discuss what the relationship is that they want between the UK and the EU?”
Mr Barnier will meet the Brexit Secretary David Davis for a fourth round of talks on Monday.
Former leader of Ukip, Nigel Farage, said Mrs May’s speech showed “a victory for Westminster and the political class”.
The speech was hailed as “positive, optimistic and dynamic” by Foreign Secretary Boris Johnson, who had published an article questioning Mrs May’s authority over Brexit.