The host of the G20 summit addressed her counterparts on Friday in a hall at the Hamburg convention centre, after video footage showed Trump shaking hands with Russian President Vladimir Putin in the first face-to-face encounter between the two men.
Angela Merkel was shown talking casually with Vladimir Putin as the leaders entered the hall, then joining French President Emmanuel Macron in a three-way discussion with President Donald Trump, who was seated between Chinese leader Xi Jinping and Britain's Theresa May.
"We all know the big global challenges and we know that time is pressing," Angela Merkel told the group.
"And so solutions can only be found if we are ready for compromise and move towards each other, but without - and I stress this - bending too much, because of course we can also state clearly when there are differences."
President Donald Trump later held bilateral talks with Mexican President Enrique Pena Nieto and Vladimir Putin.
The meeting with the Russian leader is drawing intense scrutiny because of President Donald Trump's election campaign pledge to seek a rapprochement with Moscow. So far he has been unable to deliver on that promise amid accusations from US intelligence services that Russia meddled in last year's presidential election and investigations into the President Donald Trump campaign's links to the country.
Angela Merkel, who is gearing up for a parliamentary election in September, faces the daunting task of steering the G20 towards a consensus on trade, climate change and migration - all issues that have become more contentious since Trump entered the White House half a year ago promising an "America First" approach.
Last month he pulled the United States out of a landmark international agreement aimed at combating climate change. And he is threatening to take punitive trade measures in the steel sector which would hit China, Germany, Canada and a host of other countries.
The Chinese state news agency Xinhua reported that President Xi had called on G20 nations to strengthen macroeconomic policy coordination and forestall risks in financial markets.
"Xi also urged G20 members to develop financial inclusion and green finance to make the financial sector truly drive the development of the real economy," Xinhua added.
Envoys have been working for weeks to bridge differences, and European sources said they had come up with new language on the climate issue on Thursday which would be put to the leaders for approval.
The latest draft communique sticks with language about the Paris climate accord being "irreversible" but removes a reference from an earlier version to a "global approach" that some countries felt could suggest there was a parallel track to Paris.
It also includes a new paragraph which says the United States will "work closely with other partners to help their access to and use of fossil fuels more cleanly and efficiently". Some experts were sceptical whether leaders would approve the reference to fossil fuels, which would be a clear nod to Washington.
Earlier, leaders of the BRICS countries - Brazil, Russia, India and China - called on the G20 to push for implementation of the Paris climate deal despite Trump's decision to pull out.
As the leaders met, police said they were sending reinforcements from other parts of Germany to cope with thousands of anti-capitalist protesters who set fire to cars, rubbish bins and wooden pallets in violence that Hamburg's interior minister called "frightening".
US First Lady Melania Trump was prevented from leaving her hotel to join a harbour tour because of security concerns, her spokeswoman said. German Finance Minister Wolfgang Schaeuble cancelled an appearance in downtown Hamburg.
Police said 160 officers had been injured, 45 protestors temporarily detained and another 15 taken into custody.
"We have to expect everything, and we are expecting everything," Hamburg Interior Minister Andy Grote said.
Angela Merkel chose Hamburg, the port city where she was born, to send a signal about Germany's openness to the world, including its tolerance of peaceful protests.
The summit is being held only a few hundred metres from one of Germany's most potent symbols of left-wing resistance, a former theatre called the "Rote Flora" which was taken over by anti-capitalist squatters nearly three decades ago.
On the policy front, sources said that Washington was backtracking on language condemning trade protectionism that President Donald Trump agreed to at a Group of Seven meeting in Sicily in May.
The BRICS countries pushed back in a statement saying: "We firmly support a rules-based, transparent, non-discriminatory, open and inclusive multilateral trading system, implementation and enforcement of existing WTO rules and commitments and oppose protectionism."
Hanging over the trade discussions is a threat by Washington to use a Cold War-era law to restrict steel imports based on national security concerns, a step that would hit the Chinese as well European producers.
European Commission President Jean-Claude Juncker said on Friday morning in Hamburg that the EU would respond "immediately and adequately" if the US took action on steel.
After sessions on terrorism, the global economy and climate on Friday, the leaders will be joined by their spouses for dinner at the Elbphilharmonie, a striking new glass concert hall perched atop an old warehouse building overlooking the Elbe River.
(Additional reporting by Paul Carrel, Roberta Rampton, Joseph Nasr, Sabine Siebold, Andrea Shalal, Jeff Mason; Writing by Noah Barkin; Editing by Ralph Boulton and David Stamp)
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