He stressed that the EU needed to protect its single market from Britain piggybacking on Northern Ireland's special status and to provide a "legally operational" solution - meaning he will resist a political fudge in language or a deal that would push resolution of the Irish problem until after Brexit in March and into negotiations on a future EU-UK trade agreement.
As EU leaders prepared to gather in Salzburg, Michel Barnier said he was working on an "improved" plan to avoid the return of a hard border while respecting the territorial integrity of the United Kingdom in an attempt to break the talks deadlock.
She has repeatedly vowed that she would not allow a referendum on any agreement which she reaches with Brussels over the terms of Britain's withdrawal from the EU.
But despite May's appeal at an EU summit in Austria, European Council President Donald Tusk insisted that parts of her offer are still not satisfactory more than 18 months into the negotiations and must change to keep alive hopes of concluding a Brexit deal in coming weeks.
"It was polite, it was not aggressive, she was doing her job", he told reporters as he arrived for a second day of talks.
A number of Tory eurosceptics, including Jacob Rees-Mogg, say they will not support it if it's put to MPs later this year.
In a letter to EU leaders he said they should discuss arrangements for the "final phase" of the Brexit talks "including the possibility of calling another European Council in November".
And she will not accept an EU proposal that would keep British-ruled Northern Ireland in a customs union with the bloc if there is no other plan in place to prevent a hard border across the island of Ireland.
The Democratic Unionist Party, which props up Mrs May's minority administration in Parliament, dismissed Mr Barnier's proposals as unpalatable, because they would involve a customs border between Northern Ireland and the rest of the UK.
The UK has said the EU's so-called backstop is unacceptable and its position must "evolve".
"Our proposal for the backstop on Ireland and Northern Ireland has been on the table since February", Mr Barnier said, switching from his native French to English to get his message over.
May says it will not happen because over the next couple of years, Brussels and London will work out a deal to ensure seamless UK-EU trade. "We can also clarify that most checks can take place away from the border, at the company premises or in the market".
"We have always recognised that there are unique circumstances that apply in Northern Ireland - for example in relation to checks for some agricultural products".
There was no immediate response to his comments from either Downing Street or the Department for Exiting the European Union, and it was unclear whether they would be sufficient to unlock the negotiations.
Luxembourg's Prime Minister Xavier Bettel also said time was "getting short", adding: "We both need to find a compromise".
He noted issues surrounding the Irish border and future economic cooperation.
Other EU countries are pressing for the EU to engage with British proposals, which they see as a "positive" step towards reaching a deal.
Tusk said he would call an additional summit in mid-November to seal any deal with Britain.
"If we ended up with a Chequers deal being rationalised by the Europeans I would not support that".