Under intense pressure on all sides since losing her majority, May said her task at the Brussels summit would be to set out her plans to protect EU citizens' post-Brexit rights.
But Merkel, Europe's most powerful leader, made clear that this was not at the top of her agenda as she reaffirmed Berlin's strong ties with France and its newly elected President Emmanuel Macron.
"For me the shaping of the future of the 27 is a priority coming before the issue of the negotiations with Britain on the exit," Merkel said.
Macron, attending his first summit, did not mention Brexit directly but said it was now time to get down to concrete work -- "hand in hand with Germany" -- on putting the European Union back on track after years of austerity and crisis.
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Britain's vote to leave the EU exactly a year ago on Friday was the biggest in a series of shocks that the bloc has faced, but it now insists it is turning the corner on anti-EU sentiment.
For her part, May said she would set out "clearly how the UK proposes to protect the rights of EU citizens in the UK and see the rights of UK citizens living in Europe protected."
The fate of an estimated three million Europeans living in Britain and around one million Britons living elsewhere in the EU was thrown into doubt by Britain's vote to leave the bloc last year.
May had previously refused to guarantee the rights of EU citizens in Britain until those of expatriate Britons were secured.
A European diplomat said that there was "no question of any discussion, let alone any negotiation" with May at the summit.
The issue of citizens' rights is one of three priorities in the Brexit talks which began on Monday, along with Britain's estimated 100 billion euro (88 billion pounds, USD 112 billion) divorce bill, and the fraught question of Northern Ireland, which will share Britain's only land border with the EU after Brexit.