"I think we're going to have a great trade deal", Mr Trump told interviewer Piers Morgan.
The Russian foreign ministry "liked" Trump's comments on Twitter ahead of the summit, in which Trump denounced the investigation into Russian meddling in American elections as well as previous U.S. policy.
Trump said Friday he would talk directly to Putin about Russia's meddling in the 2016 election, but mockingly told reporters it is their favorite topic and that he doubted he would produce a "Perry Mason" moment in which Putin admitted to being involved.
Trump heads into the summit with little clear agenda other than to strengthen his personal rapport with Putin, which he thinks is crucial to improving relations between Washington and Moscow.
"Well, I might. I hadn't thought of that, but, certainly, I will be asking about it", Trump said.
The two are scheduled to hold a joint press conference later in the day on Monday. He said,"I would like to congratulate you on a very great world cup, one of the best ever". The summit also started almost an hour late.
Trump landed in the capital city, Helsinki, late Sunday.
Then Trump complained once again that President Obama did nothing to stop the Russian election meddling that he publicly encouraged.
"Our relationship with Russian Federation has NEVER been worse thanks to many years of United States foolishness and stupidity and now, the Rigged Witch Hunt!" wrote Trump.
Finland has a long legacy of hosting US-Soviet and US-Russian summits because of its geographic location and perceived neutrality.
Putin also addressed the issue of "so-called interference", saying he "had to repeat that the Russian state never interfered, and does not plan to interfere in internal American electoral process".
Following the one-on-one meeting, Trump and Putin began a larger meeting with other US and Russian officials, and will end the summit with a joint press conference.
"That's what one of the purposes of this meeting is, so the president can see eye-to-eye with President Putin and ask him about it", Bolton said.
The indictment raised pressure on Trump to raise Moscow's so-called "malign activities" with Putin.
The spokesman said the Government remained committed to "constructive dialogue" with Washington over steel tariffs, after Mr Trump described the European Union as "a foe" of the U.S. in trade terms. Also the Federal Bureau of Investigation is our foe, President Obama is our foe. Those agents broke into Democratic Party email servers, stole tens of thousands of messages, and leaked them on Russian-controlled web sites. Obama ordered an intelligence investigation and spoke to Putin about what the USA government determined Russian Federation was up to.