The two leaders begin closed-door talks in a much-anticipated summit held in the Finnish capital Helsinki.
Putin told Trump at the start of their summit on Monday that “the time has come to talk thoroughly about bilateral relations, as well as various hotspots in the world”.
For his part, Trump first congratulated Putin with the successful organisation of “one of the best World Cups of all times”.
He said their discussions would involve trade, the military, missiles, nuclear weapons and China, including their “mutual friend” China’s Xi Jinping.
Trump also said that “getting along with Russia is a good thing, not a bad thing”.
“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 alleged Russian meddling in the US presidential campaign. 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.
But Trump has insisted it is “a good thing to meet”.
Last week, US authorities indicted 12 Russian military intelligence officers for alleged interference in the 2016 presidential election.
Al Jazeera’s James Bays, reporting from Helsinki, said Trump failed to highlight the issues that seemed to be viewed as the most important in his country.
“There are issues that people at home what Trump to press President Putin on,” Bays said.
“Those are all very important global issues, but he didn’t mention them,” he added.
“He didn’t mention [the Russian] election meddling in the US, cybercrime during the 2016 election and the indictment of 12 Russians in the last few days by the special counsel,” he said.
“Neither did he mention the Russian meddling in Ukraine and the annexation of Crimea. And he didn’t mention the Middle East, particularly Syria.”
Before Monday’s meeting, Trump criticised previous US administrations on Twitter, saying: “Our relationship with Russia has NEVER been worse thanks to many years of U.S. foolishness and stupidity and now, the Rigged Witch Hunt!”
The Russian foreign ministry “liked” his tweet.
The Kremlin had previously said it did not expect much from the meeting but hoped it would be a “first step” to resolving a crisis in ties.
“Presidents Trump and Putin respect each other and they get along well,” said Dmitry Peskov, the Kremlin spokesman. “There is no clear agenda. It will be determined by the heads of state themselves as they go along.”
Critics and Trump’s own advisers have urged Trump to use the summit to press Putin hard about “malign” activities.
SOURCE: AL JAZEERA AND NEWS AGENCIES