The DUP leader accused Leo Varadkar's government of "using the negotiations to put forward their views on what they believe the island of Ireland should look like in the future".
An EU official said that May agreed in the one-hour discussions that December 4 was the "absolute deadline" to allow the EU's Brexit negotiator Michel Barnier to recommend moving onto the next stage on trade and future ties.
The EU will decide whether sufficient progress has been reached to move on to phase two in Brexit talks after debating the latest steps by the United Kingdom government at a December summit.
"The UK will need to give credible assurances as to how to avoid a hard border before December 4, as it is still unclear how this can be done", the official said.
Asked by Sky News if his country would be willing and able to veto talks moving to the trade phase, and if he has the support of the rest of the European Union, the Irish Foreign Minister, Simon Coveney was firm.
"We can't move to phase two [of the negotiations] based on aspirations, but on the basis of a credible road map or the parameters around which we can design a credible road map", he said.
In an interview with BBC Radio Ulster's Inside Politics show, she added: "But we are not there yet, let's wait and see what happens in Dublin over the coming days".
"I hope and expect that we can get that by December so that we can all move on".
"I don't think anyone wants a general election".
Coveney met with European Union chief negotiator Michel Barnier, who reaffirmed the other 26 member states' support for Ireland's position.
"What we are talking about is whether we can move on to opening up phase two in parallel with phase one issues in December".
"The Irish issue is very worrying".
"These negotiations are continuing but what I am clear about is that we must step forward together", she said.
"The UK will need to give credible assurances as to how to avoid a hard border before 4 December, as it is still unclear how this can be done", the European Union source said.
"Ministers must change course and stop recklessly ruling this out as an option".
The meeting with the German Chancellor Angela Merkel was hastily organised and an important moment of engagement.
Assessing his meeting with Mrs May, the Danish Prime Minister, Lars Lokke Rasmussen, told Sky News: "It was very constructive".
"Ministers are in such chaos because their position is so impossible and illogical. We want to ensure that movement of people and trade across that border can carry on as now".
"The circle can not be squared".
Tusk met with British prime minister Theresa May after the EU's Eastern Partnership summit.
Discussing allegations of Russian interference in Western elections, the European Council president incorrectly suggested that Mrs May had accused Moscow of "hostile activities during the British referendum". It wasn't a negotiating meeting.
"Certainly in respect of the Stormont talks I assume that if an election as called in the Republic of Ireland that Sinn Fein's focus will be entirely on Dublin and the elections down there", said Mrs Foster.