If Trump's instinct is right and he finds common ground with Putin, then the pair's Helsinki Summit may take the heat out of some of the world's most dangerous conflicts.
But the Washington-Moscow rivalry has a long history and there are there many points of friction that could yet spoil Trump's hoped-for beautiful friendship.
With the foes at loggerheads over Syria, Ukraine, pipeline policy, espionage and election interference, even Trump cautioned: "I'm not going with high expectations."
The brash billionaire property magnate has been president for 18 months, while the 65-year-old former KGB officer has run Russia for the past 18 years.
The 72-year-old president nevertheless has a high opinion of his ability to woo tough opponents, such as North Korea's Kim Jong Un, whom he met at a summit last month.
"I think it's a good thing to meet. I do believe in meetings," Trump insisted in an interview with CBS News that aired before he touched down in Helsinki.
In the same interview, Trump admitted that Russia remains a foe, but he put Moscow on a par with China and the European Union as economic and diplomatic rivals.
The Kremlin has also played down hopes that the odd couple will emerge from their first formal one-on-one summit having resolved the issues poisoning relations.
Putin, who played host at the World Cup final in Moscow on Sunday and was due to arrive in Finland later Monday, has remained terse in the run up to the summit.
But on Friday his adviser Yuri Ushakov also played down expectations, saying: "The state of bilateral relations is very bad.... We have to start to set them right."
Giving up ground?
Indeed, after a week in which Trump threatened to up-end a summit of the NATO allies, anxious European leaders may be relieved if not much comes of the Helsinki meeting.
Many fear that Trump -- in his eagerness to prove that he was right to seek the summit with Putin despite Russia's often hostile stance -- may give up too much ground.
In the run-up to talks, Trump has refused to personally commit to the US refusal to recognise Russia's annexation of Crimea, leaving open the possibility of a climb-down.
If Washington were to de facto accept Russia's 2014 land-grab, this would break with decades of US policy send tremors through NATO's exposed eastern flank.
Trump's critics in Washington will be watching this -- and also how he handles the growing evidence that Russian agents intervened in America's 2016 presidential race.
Last week U.S. special prosecutor Robert Mueller indicted 12 more Russian intelligence officers for allegedly hacking Trump rival Hillary Clinton's computer server.
There will be outrage at home if Trump does not confront Putin over the scandal, but the mercurial U.S. leader would not say whether he would demand the suspects' extradition.
"Well, I might. I hadn't thought of that. But I certainly.... I'll be asking about it," Trump told CBS.
Senior diplomat and now chairman of the Council on Foreign Relations Richard Haass said that for centuries world order has depended on "non-interference in the internal affairs of others and respect for sovereignty."
"Russia has violated this norm by seizing Crimea and by interfering in the 2016 US election. We must deal with Putin's Russia as the rogue state it is," he tweeted.
But of all the topics that may come up in the meetings it is Syria that may prove most important.
Despite the doubts of his top national security advisers, Trump is keen to withdraw US troops from eastern Syria, where they have been battling the Islamic State.
Reports suggest he may seek a deal that Russia work with Israel to contain Iran's influence, in exchange for allowing Putin's ally Bashar al-Assad to stay in power.
This could free up U.S. troops to withdraw, but would also -- as with Crimea -- mark a major victory for Putin and a betrayal of local U.S. allies on the ground.
'Not good enough'
Trump certainly does not expect to win over the Washington media and foreign policy establishment.
In a bitter tweet sent as he flew between his golf resort in Scotland to the Finnish capital, he said even if he was handed the keys to Moscow "it would not be good enough."
Helsinki may not be impressed either.
On the eve of the summit, more than 2,000 people denounced Trump and Putin and hailed human rights, press freedom and dissent as they marched in the city's Senate Square.