Trump, bent on forging a personal bond with the Kremlin chief despite allegations of Russian meddling in US politics, went into the summit blaming "stupidity" by his predecessors for plunging ties to their present low.
Looking sombre, the two leaders exchanged a few opening remarks in front of the press at the start of their summit in Helsinki.
Putin, basking in congratulations from Trump and other world leaders for the successful staging of the World Cup in Russia, said: "The time has come to talk in a substantive way about our relations and problem areas of the world."
"Frankly, we have not been getting along for the last number of years. And I really think the world wants to see us get along. We are the two great nuclear powers," he said.
"I've not been here too long (as president), it's getting close to two years, but we'll be having an extraordinary relationship, I hope so."
Shortly before the summit opened, Trump was asked if he would press Putin over Russia's alleged manipulation of the 2016 election that brought the mercurial property tycoon to power. He said only: "We'll do just fine."
Many US critics had called for the summit's cancellation after new revelations surrounding the alleged election meddling.
If the pair do find common ground, the summit may take the heat out of some of the world's most dangerous conflicts, including Syria.
But there are many points of friction that could yet spoil Trump's hoped-for friendship with the former KGB spymaster.
Trump began the day by firing a Twitter broadside at his domestic opponents, blaming the diplomatic chill on the investigation into alleged Russian election meddling.
Russia's foreign ministry tweeted in response: "We agree." Trump's US opponents tried, in turn, to gain traction for the hashtag #BAF (Blame America First).
But over breakfast with Finland's President Sauli Niinisto, he insisted NATO "has never been stronger" and "never been more together" thanks to his insistence on all allies paying their fair share.
The Kremlin has also played down hopes that the odd couple will emerge from their first formal one-on-one summit with a breakthrough.
On Friday Putin's adviser Yuri Ushakov said: "The state of bilateral relations is very bad.... We have to start to set them right."
Those leaders are already fuming over Trump's imposition of trade tariffs on various countries, including Russia.
Turning the tables, European Union President Donald Tusk said Trump was guilty of "spreading fake news" with his remark about foes, and warned that the trade tensions could spiral into violent "conflict and chaos".
"Europe and China, America and Russia, today in Beijing and in Helsinki, are jointly responsible for improving the world order, not for destroying it," he tweeted.
"I hope this message reaches Helsinki."
Protesters have been on the streets of Helsinki to denounce the policies of both Trump and Putin. Greenpeace draped a giant banner down a church tower urging: "Warm our hearts not our planet."
Trump is also under pressure from Britain to press Putin over the nerve agent poisoning of four people in southern England. (AFP) ZH
(This story has not been edited by Devdiscourse staff and is auto-generated from a syndicated feed.)