The government says it is "optimistic" about an Irish border agreement as pressure mounts to find a Brexit deal.
The DUP said the proposals, which aimed to avoid a "hard border" by aligning regulations on both sides, were not acceptable.
At the summit, European leaders will decide whether enough progress has been made in the negotiations so far to open trade talks.
"The Commission is satisfied that sufficient progress has been achieved in each of the three priority areas", the European Commission said in a statement.
Commission chief Jean-Claude Juncker's chief of staff Martin Selmayr tweeted a picture of white smoke - the sign used by the Vatican to signify the election of a new pope - shortly after May's arrival.
Amid signals Britain could seek "regulatory alignment" with the European Union to ensure there does not need to be a hard border, Foreign Secretary Boris Johnson insisted any Brexit deal must mean "taking back control of our laws, of our borders and of our cash".
The party wants Northern Ireland to leave the European Union on the same terms as the rest of the UK.
Juncker told reporters Friday that "I believe that we have now made the breakthrough that we needed".
The EU says May has an effective deadline of Sunday night if she wants to seal a deal and hope to have agreement on trade talks in time for the EU summit on December 14-15.
On Wednesday the Republic of Ireland's Prime Minister Leo Varadkar said he expected Theresa May to come up with a new wording aimed at satisfying all parties, adding: "I expressed my willingness to consider that".
Foster said there is "still work to be done in London" after her short telephone call to the prime minister on Wednesday morning over the Irish border issue in the Brexit talks.
Q. What happens if Mrs May gets agreement on plans to maintain a soft Irish border?
The MPs included former cabinet ministers Stephen Crabb, Dominic Grieve, Anna Soubry and Nicky Morgan.
Phase two of the Brexit talks will focus on the transitional arrangements that will kick in after the UK formally leaves the EU in 2019 and the future EU-UK trading relationship.
In Brussels, Mr Barnier agreed that EU ambassadors must discuss any proposed new agreement at a meeting on Friday evening, if it is to be on the table at the EU Council next Thursday.
Scotland's nationalist leader showed little patience, accusing the British government of being "totally and utterly incompetent" on Brexit.
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