Update 2.30pm: The Taoiseach has suggested the British government hasn't thought Brexit through.
Ahead of his talks with Mrs May, 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."
His comments came after what was described as 'a fresh clash with Theresa May over Brexit' in which he demanded a commitment that there will be no return to a "hard border" between Northern Ireland and the Republic.
Attending a European summit in Sweden, the Taoiseach made clear the Brexit negotiations could not move on to the second phase until the future status of the border was clear.
Mrs May is anxious to secure the agreement of EU leaders to open discussions on Britain's future relations with the bloc - including a free trade deal - when they meet next month in Brussels.
However Mr Varadkar, who met Mrs May in the margins of the gathering in Gothenburg, said that would require further "concessions" from the UK.
"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," he told reporters.
"If we have to wait until the New Year, if we have to wait for further concessions, so be it."
Update 11.30am: Leo Varadkar and Theresa May have clashed over one of the key sticking points in the Brexit process.
The Taoiseach suggested that the British Prime Minister and Tory Brexiteers had not thought through their approach to the Irish border issue.
Mr Varadkar called for commitment that there will not be a hard border on the island of Ireland after the UK leaves the European Union.
And he indicated that a form of words would need to be found for that guarantee before Brexit talks could be allowed to move on to phase two, discussing a possible trade deal.
The three issues where "sufficient progress" must be made before the EU will agree to move on to trade talks include the financial settlement, citizens' rights and the border between Northern Ireland and Ireland.
Ahead of the face-to-face talks with Ms May on the margins of a European summit in Gothenburg, 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."
He told reporters: "Britain having unilaterally taken the customs union and single market off the table, 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."
The Taoiseach said it was "certainly possible" that trade talks could be given the go-ahead next month "but if we have to wait until the New Year, if we have to wait for further concessions, so be it".
Downing Street said the talks between Mr Varadkar and Ms May on Brexit had been "constructive".
But a source described the meeting as a "solid exchange of views" and said Mr Varadkar was "optimistic but slightly frustrated in terms of we are so close but so far, really".
The source said: "We just want a clear commitment, a clear form of words that points to a solution with regards to the Irish situation that would allow us to move on to phase two."
Ireland "cannot make a leap in the dark", the source added.
A British Government source said there was "work to do" on the issue and "everyone's very honest about that".
European Council president Donald Tusk will also deliver a tough message to Ms May when they hold talks in the margins of a summit in Gothenburg, telling her that progress on to the next phase of Brexit talks cannot be taken for granted.
The Prime Minister said she hoped the European Union would respond "positively" to her efforts to secure progress ahead of December's European Council meeting, where leaders will consider the UK's call for trade talks to begin.
A Brussels source said: "Mr Tusk will inform Ms May that such a positive scenario is not a given, it will require more work and that time is short.
"And he will ask Ms May how the UK plans to progress on the three key issues for phase one."
Ms May repeated her promise that the UK will "honour our commitments" amid speculation she is prepared to increase the amount she is prepared to pay in the divorce settlement with the EU.
She said: "We look forward to the December European Council. We are continuing to look through the issues.
"I was clear in my speech in Florence that we will honour our commitments.
"But of course we want to move forward together, talking about the trade issues and trade partnership for the future.
"I have set out a vision for that economic partnership, I look forward to the European Union responding positively to that so we can move forward together and ensure that we can get the best possible arrangements for the future that will be good for people in the United Kingdom and across the remaining EU27."
The Prime Minister held talks with her Swedish counterpart Stefan Lofven on Thursday night. He said the UK needs to clarify what the financial settlement would cover, and it was "very difficult to say" whether trade talks would be given the go-ahead in December.
If leaders do not agree to move to the second phase at the summit in Brussels on December 14-15, then it could mean no progress until the next scheduled European Council in March.
That would add to business uncertainty and increase the potential for the UK to leave without a Brexit deal. Britain's Brexit Secretary David Davis said many EU nations were now keen to move on to trade talks, but notably omitted Germany and France from his list.
Speaking to the BBC in Berlin this morning, Mr Davis said: "Many of them do want to move on.
"They see it as very important to them. Countries like Denmark, countries like Holland, countries like Italy and Spain, countries like Poland can see the big, big benefits in the future deal we are talking about."
The Gothenburg social summit brings together political leaders and other key players to discuss a new European Pillar of Social Rights for workers' rights.
Update: 6.30am: Brexit will top the agenda, when the Taoiseach and the British Prime Minister Theresa May hold a face to face meeting in Sweden today.
Leo Varadkar is expected to tell Theresa May that he won’t accept a hard border under any circumstances.
It will be the first time the pair have met since a document emerged from Europe suggesting the North stay in the EU after Brexit.
Sinn Féin MEP Liadh Ní Riada says it’s crucial Ireland stands firm in talks.
"I think it’s high time that the Irish government has to have key people at that negotiation table. They cannot afford to be commentators or spectators on something that is so key to Ireland’s future," Ms Ní Riada said.
The meeting today takes place ahead of the Social Summit 2017 in Gothenburg.