Trump's "America First" approach and dismissive remarks about the EU and North Atlantic Treaty Organisation have and bolstered calls for the European Union to take on an increased defence role, while his decision to pull out of the Paris climate pact infuriated Europe.
Reflecting confusion on the continent about what kind of Brexit she will ask for, summit chair Donald Tusk said ahead of a separate meeting with May: "We can hear different predictions, coming from different people, about the possible outcome of these negotiations: hard Brexit, soft Brexit or no deal".
Tusk made the comments a few hours before a bilateral meeting with British Prime Minister Theresa May, quoting John Lennon's song "Imagine": "So who knows?"
The UK election has left Mrs May seriously weakened, heading a minority government, which has fuelled uncertainty about the whole Brexit process.
Many EU leaders assume Brexit is now inevitable and see talk of a U-turn, which would need EU consent, as a distraction.
May has repeatedly insisted she is not willing to guarantee their future without a reciprocal pledge about the future of British citizens living in other European Union countries.
Mrs May is expected to urge member states to follow the lead of a UK-French agreement to put pressure on internet companies to remove extremist material from their sites, sealed with President Emmanuel Macron on a visit to Paris last week.
"It will be a brand new door, to a new Europe", he added. "It's not up to me to say it's closed", he told The Guardian.
Theresa May's personal position is also very weak, following a disastrous election campaign where she refused to participate in leader's debates and tended to appear before vetted sections of the public in highly managed events.
Similarly, Britain's Secretary of State for Exiting the EU David Davis said he was "encouraged by the constructive approach that both sides have taken".
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"The UK's position represents a fair and serious offer and one aimed at giving as much certainty as possible to citizens who have settled in the UK", May told her colleagues.
The EU has made a priority of the rights of three million European citizens living in Britain, plus a million Britons resident in Europe.
"That's been an important issue".
EU Chief Negotiator Michel Barnier said that Britain and the European Union have agreed on the priorities and the agenda of Brexit talks after the first round that began on Monday. "That work is starting".
She should, Soros said, seek to keep Britain in the EU's single market as Britain tried to extract itself.
Tusk meanwhile insisted the remaining 27 members had a renewed sense of optimism about the bloc's future after years of crisis and mounting anti-EU sentiment culminating in the Brexit vote.
Britain's political instability has caused concerns in European capitals hoping to get the negotiations over with quickly and cleanly, as has May's oft-repeated threat to walk away from the talks without a deal if necessary.
United Kingdom opposition parties had urged the government to make a unilateral guarantee to the European Union migrants - but ministers have insisted a reciprocal deal is needed to ensure British expats are protected.
"Anything that provides a great deal of security for people living in Britain or planning to live in Britain during the time in which Britain is still a member of the European Union would be useful", she said, underlining European Union attachment to a cut-off only in 2019. "But the clear focus must be on the future of the 27, so that we have the best results".
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