Barnier highlighted two key areas of disagreement - on the legal protection for the rights of the 3 million European Union citizens in Britain and on a pre-Brexit settlement from Britain for a share of outstanding financial commitments made during its membership.
Prime Minister May proposed in a speech in Florence last week a two-year transition - or "implementation" - period for the country after Brexit, during which it will still adhere to the bloc's rules and regulations.
At a joint press conference in Brussels after their latest round of talks, Brexit secretary David Davis insisted they had "made important progress and capitalised on the momentum created by the prime minister's speech".
He said a solution was also needed on the border with Ireland which fully respected both requirements of the European Union single market and the Good Friday Agreement.
The resolution said the European Parliament "is of the opinion that in the fourth round of negotiations sufficient progress has not yet been made on citizens' rights, Ireland and the Northern Ireland, and the settlement of the United Kingdom's financial obligations".
Mrs May's charm offensive came after the fourth round of Brexit talks that began in Brussels on Monday made little progress.
"Our EU colleagues in stores and supply chains are part of the reason that British retailers are often able to deliver affordable and high-quality goods", she said. "Further work is needed in coming weeks and coming months".
A sticking point remains the future role of the European Court of Justice, something Britain doesn't want controlling our laws anymore.
But the Union continues to demand that people also have the right to pursue grievances at the EU's own court in Luxembourg - a red line for a British government keen to show voters who backed Brexit previous year that they have taken back control.
'We are working quickly through a number of complex issues, but there remain some points where further discussion and pragmatism will be required to reach an agreement'.
"Thanks to the constructive and determined manner in which both sides have conducted these negotiations I believe we are making decisive steps forward", Davis said.
Ms Fairbairn and Ms O'Grady on Thursday said however, that any firm guarantee on the movement of citizens must be implemented independently from the rest of the negotiations, to "avoid the risk that "no deal" in March 2019 leads to uncertainty and heartache for millions of people". "We are getting closer to the UK's withdrawal and I think that this moment should be a moment of clarity", he said.
On the financial settlement he repeated May's message from Florence that the remaining 27 European Union countries will not have "to pay more or receive less over the remainder of the current European Union budget plan" once Britain is outside the bloc.
- Morning Scan Equifax CEO Smith 'retires'; Senators scold SEC chief
- NFL players should not protest during the national anthem: Fmr. Patriots owner
- Thousands Flee Bali's Mount Agung After Volcano Threat Level Is Raised
- It's 2017 and Saudi Arabia Will Finally Allow Women to Drive
- Rihanna criticises Donald Trump's response to Puerto Rican hurricane crisis
- Aaron Judge Breaks MLB Rookie Record With 50th Home Run
- Defoe strengthens Euros case
- Top smartphones for gaming
- Guardiola uneasy over lofty Manchester City comparisons
- Uber's license to operate in London won't be renewed