Tusk was "more optimistic" after meeting May a week ago on the sidelines of an European Union summit in the Swedish port city of Gothenburg, but when it came to tabling a new offer "the earlier the better", an European Union source told AFP.
One potential sticking point remains the fact that May's cabinet says that an increased offer can only be part of a final deal on leaving the European Union - which is at odds with the Brussels line on tying up divorce terms before discussing future relations.Ireland is meanwhile pushing to ensure its concerns about the border with British-ruled Northern Ireland are taken into account, adding an unexpected hurdle to a December deal. Does it want to stay close to the European model or does it want to gradually move away from it? "The UK's reply to these questions will be very important, and even decisive", he said. "I do not say this to create problems but to avoid problems".
Speaking to a Brussels audience at a Centre for European Reform conference, Barnier said the United Kingdom would have access to the single market, but stressed this was not equal to membership. We've seen some good and positive decisions for the United Kingdom, but they were taken prior to the referendum.
British firms, he said, would lose their financial service passports, which enable banks and institutions in the City of London to reach a market of 500 million consumers and 22m businesses.
The EU negotiator added the bloc will be ready to offer the "most ambitious" free trade agreement.
The talks are hoped to pave the way for meetings in December with EU Brexit negotiator Michel Barnier, and European Commission president Jean-Claude Juncker, who himself said this week that the EU and the United Kingdom had got through the toughest negotiations.
Britain wants EU leaders meeting at a summit in December to agree that "sufficient progress" has been made on the divorce bill, the issue of Ireland (Other OTC: IRLD - news) and EU citizens' rights, so that talks can move onto a future UK-EU trade deal. "Everyone is talking to everyone, already, at all levels", he said. "It is quite remarkable that the Good Friday agreement is being taken more seriously by French, Italian and German politicians than it is by British politicians", he said.