May and European Commission President Jean-Claude Juncker fell short of a breakthrough, despite some encouraging progress on the Irish border issue.
Ms Foster had issued a statement warning it would not accept "any form of regulatory divergence which separates Northern Ireland economically or politically from the rest of the UK".
May and Juncker made no comment to reporters when they met at the European Union executive's Berlaymont headquarters for a lunch that diplomats and officials hope can seal a breakthrough that would open the way to negotiations on future trade relations. "It was reported as if it was true, and now it turns out it was propaganda from the Irish Government", he said.
This week began with the DUP leader, Arlene Foster, reminding the EU, Irish Taoiseach Leo Varadkar and Theresa May that, whether they like it or not, her party holds some cards in the Brexit game. The lack of progress so far has raised concerns that Britain may not have a deal by the time it officially leaves on March 29, 2019.
Mr Juncker insisted that a deal could still be reached this week, having described Mrs May as a "tough negotiator". "But on a couple of issues some differences do remain which require further negotiation and consultation".
The EU and the United Kingdom are nearing agreement on some divorce terms, including the size of the bill that Britain must pay as it leaves and the rights of citizens affected by Brexit.
Such a move would essentially mean Northern Ireland remained inside the Customs Union and the Single Market, leading to a potential decoupling from the rest of the UK.
It is understood this is a sticking point in reaching a deal on post-Brexit citizens' rights, the part of the phase one divorce negotiations that had previously been considered to be the most advanced.
May depends in parliament on a pro-British party in Northern Ireland that rejects any deal which would divide the province from the British mainland.
But that task has been complicated by the DUP's show of strength, the publication of the sensitive wording from the draft negotiating text and Leo Varadkar's public insistence that the British government had signed up to a formula which the DUP found so hard to swallow.
He added: "I still hope that this matter can be concluded in the coming days as agreed".
"It is now getting very tight but agreement at December (summit) is still possible", he tweeted.