In Germany, Stefanie Bolzen and Hannelore Crolly of the centre-right Die Welt say events have "taken a unsafe turn" for Mrs May, despite EU Commission President Jean-Claude Juncker's insistence that the talks were "not a failure".
Katya Adler said there was an "upbeat mood" of "cautious optimism".
Ireland has called on Britain to provide details of how it will ensure there is no "regulatory divergence" after Brexit in March 2019 that would require physical border controls.
But, they warn, other member states, including powerhouses Germany and France, are growing nervous that they will not have enough time to scrutinize draft guidelines for the trade negotiations if they do not receive them a week before a summit next Friday at which leaders would approve them.
"We're not quite yet where we need to be".
May will hold talks with top European Union officials later Wednesday. A British spokesman said: "With plenty of discussions still to go, Monday will be an important staging post on the road to the crucial December Council".
The UK voted for Brexit past year and is due to leave in March 2019, but negotiations between the European Union and the UK have not yet reached a breakthrough.
According to a agency report, deals were reached this weekend on the United Kingdom "divorce bill" and citizens' rights.
"We are at the point of progressing to the next stage", Mrs May told parliament, after Northern Ireland's Democratic Unionist Party (DUP) blocked a proposed agreement on the Irish border being discussed in Brussels on Monday.
"We will do that while we respect the constitutional integrity of the United Kingdom and while we respect the internal market and protect the internal market of the United Kingdom", May said. Irish Deputy Prime Minister Simon Coveney said on Sunday his country had "no desire" to delay the UK's Brexit talks, although not enough progress had been made so far.
Colm Kelpie, who writes on Brexit for the Irish Independent, says Mrs May has "little wriggle room" in her talks with the Democratic Unionists, and expresses amazement that the "DUP was not already on board if the wording of a text was agreed that was acceptable to the Irish Government".
"Hopefully we'll find a way forward today", he said. "But at this stage it is very hard to make a prediction".
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