May spokesman James Slack said she told the ministers she was confident of reaching a deal, but "while the United Kingdom should aim to secure a withdrawal agreement as soon as possible, this should not be done at any cost".
According to the Express, Mr Barnier told reporters Tuesday: "Today, we're not there yet".
Whitehall sources stressed that there was "much work to do" on what was a "fabulously complicated" solution to the problem of keeping open the border of Northern Ireland with Ireland post-Brexit.
Britain and the European Union both wish to keep the border between EU-member Ireland and Northern Ireland in the United Kingdom open after Brexit, as it is seen as crucial to the 1998 Good Friday peace accord that ended decades of sectarian bloodshed in Northern Ireland.
Mrs May has been boosted by Mr Varadkar's willingness to negotiate a review mechanism, thought to be part of proposals for the United Kingdom to enter an effective customs union with the EU under a backstop arrangement to prevent a hard Irish border. Britain's desire to leave the customs union is not easily reconciled with preserving the integrity of the EU's single market.
Although Mrs May has stated 95% of the UK's withdrawal agreement is settled, the Irish border issue continues to prove a sticking point in exit talks.
"Backstop means backstop. And a backstop can not have a time limit". Barnier said the backstop can have no end-date. Without an operational backstop there will not be an accord and there will not be a transition period.
Prime Minister Theresa May has previously stated she will not consider any proposals that would see Northern Ireland "carved out" of the United Kingdom.
Mr Donaldson said a no deal would have "serious consequences" for the Republic of Ireland's economy and that the United Kingdom "won't have to pay a penny more" to the EU.
With Britain due to leave the European Union on March 29, May is running out of time to find an agreement that is acceptable both to her divided Conservative government and to fellow European Union nations.
More than 70 business leaders - from lastminute.com founder Martha Lane Fox and former J. Sainsbury PLC chief Justin King to Cobra Beer founder Karan Bilimoria and ex-chairman of Rolls-Royce Holdings PLC Simon Robertson - signed a letter arguing that both the government's current plans for Brexit, and a no-deal Brexit, would be bad for companies and jobs.
In response, Sir Jeffrey said Dublin's stance was making a no-deal Brexit likely.
Expectations are rising that United Kingdom negotiator Olly Robbins will be pressing hard to finalise a deal in Brussels this week, to set the scene for a special Brexit summit later in the month to secure the approval of the leaders of the 27 remaining member states.
But he rejected the idea Britain could unilaterally end the arrangement, something Brexit Secretary Dominic Raab and other eurosceptics in her Conservative party want.