Brussels' chief Brexit negotiator has told Theresa May to produce "realistic and workable" proposals as the Prime Minister called for the process to accelerate.
Michel Barnier said "huge and serious divergence" remains over issues relating to Ireland and Northern Ireland and called for an extra session of negotiations with UK officials on Monday.
The Prime Minister said she hoped a new phase in the Brexit talks would be possible after the publication of the Government's White Paper calling for negotiations to speed up and intensify once the document is published.
Details of the White Paper are expected to be agreed by Cabinet ministers at next Friday's Chequers away day.
Mrs May's participation in the European Council summit ended in the early hours of Friday morning after a marathon session on proposals to address the migrant crisis.
She warned EU leaders the safety of their citizens would be put at risk if a deal on security co-operation after Brexit is not reached.
Leaving the summit in Brussels, she said: "We are going to be publishing our White Paper shortly and I want to see the negotiations accelerating and intensifying thereafter."
She urged EU leaders to change the mandate given to their negotiators in order to allow UK involvement in key law enforcement initiatives including the Prum mechanism for sharing DNA profiles and other data, the Second Generation Schengen Information System (SIS II) database of "real-time" alerts about individuals interest to EU law enforcement and the European Criminal Records Information System (ECRIS).
Without a deal she told her counterparts "we would no longer be able to share real-time alerts for wanted persons, including serious criminals" and "our collective ability to map terrorist networks across Europe and bring those responsible to justice would be reduced".
The remaining 27 EU leaders were continuing their discussions on the future relationship between the UK and Brussels in Mrs May's absence.
Mr Barnier said the EU wanted an "ambitious" partnership but it had to be based "on our values and our principles".
Arriving at the summit he said: "We have made progress but huge and serious divergence remains, in particular on Ireland and Northern Ireland.
"Now we are waiting for the UK White Paper and I hope it will contain workable and realistic proposals.
"The time is very short. We want a deal and are working for a deal, the time is short and I am ready to invite the UK delegation to come back to Brussels next Monday."
The leaders of the EU 27 were returning to the summit following a stormy working dinner on Thursday evening which stretched into the early hours of Friday as desperate attempts were made to find a compromise on migration.
Confirmation that a deal had been reached only came shortly after 4.30am local time when European Council president Donald Tusk tweeted conclusions had been agreed.
The issue has assumed greater political significance in Europe because of the threat to German chancellor Angela Merkel's position over her approach to letting migrants settle in her country.
Meanwhile Italy's Giuseppe Conte used the summit to demand a fundamental change in the bloc's migration policy, saying his country received little help even though it was at the forefront of receiving migrants from across the Mediterranean.
Speaking shortly after 5am in Brussels, Mrs May acknowledged there had been "lengthy discussions" on the EU's approach.
She said: "We have come to positive conclusions, a lot of them around what the United Kingdom has been encouraging for some time, which is taking more action upstream in countries of origin so that we can ensure that people aren't having to make and aren't making these very dangerous journeys, often travelling many miles, often at the hands of the people smugglers and making the dangerous trips across the Mediterranean where we still see some people dying."
Irish Taoiseach Leo Varadkar said there should be a degree of "burden sharing" on the transfer of migrants to other countries.
He added: "The 28 members agreed a compromise framework on managing migration largely based on the principle that this is a European problem and one that we need to work together on."
EU diplomats said the leaders finally found agreement on a vaguely worded concept centring on reception centres to deal with migrants and asylum seekers in EU nations which would volunteer to have them.
Mr Tusk had pushed the idea of "regional disembarkation platforms" - facilities outside the EU where migrants picked up by search and rescue missions would be assessed to decide whether they are refugees or risking the Mediterranean crossing for economic reasons.
Within EU territory, those who are saved could be taken to "controlled centres" set up in member states for processing.