In a clear signal that Brussels is holding out for more cash, the top Eurocrat told the Prime Minister “much more progress” was needed by the beginning of next month for trade discussions to begin.
The pair met on the side lines of a European summit on social policy in Gothenburg, Sweden.
Mr Tusk said after meeting Mrs May: “We will be ready to move on to the second phase already in December, but in order to do that we need to see more progress from the UK side.
“While good progress on citizens’ rights is being made, we need to see much more progress on Ireland and on the financial settlement.”
A summit of EU leaders in Brussels on December 14-15 is set to decide whether to give the green light to the European Commission to allow the talks between the British and EU negotiating teams to begin thrashing out a long-term trade deal.
Mr Tusk said he and Mrs May agreed to meet again next Friday “to assess the situation in more detail”.
Mrs May told reporters as she left Gothenburg: “We are agreed that good progress has been made but there is more to be done, that we should move forwards together towards that point where sufficient progress can be declared and we can look ahead to what I have already said I want to see as a deep and comprehensive and special partnership between the UK and the remaining 27 members of the EU.”
Mrs May also met French President Emmanuel Macron and Irish prime minister Leo Varadkar in Gothenburg.
But the latter yesterday raised the stakes in the Brexit wrangling by suggesting his country was ready to delay the start of trade talks beyond the start of next year unless the UK offers further concessions on the UK-Ireland border.
After meeting Mrs May in the margins of the summit in Gothenburg, he told reporters: “Before we move to phase two talks on trade, we want taken off the table any suggestion that there will be a physical border, a hard border, new barriers to trade on the island of Ireland.
“If we have to wait until the New Year, if we have to wait for further concessions, so be it.”
In a scathing assessment of the UK’s preparedness for the complexity of Brexit negotiations, Mr Varadkar said: “It’s 18 months since the referendum, it’s 10 years since people who wanted a referendum started agitating for one.
“Sometimes it doesn’t seem like they have thought all this through.”