Putin told the media that the two leaders had kept in regular touch over telephone and in meetings in multilateral scenarios and it was time to have an in-depth discussion of bilateral ties and urgent international affairs, reports Xinhua news agency.
Trump said he would raise issues related to "trade, to military, to missiles, to nuclear, to China."
"I think we have great opportunities together as two countries that frankly, we have not been getting along very well for the last number of years," he said.
Trump praised Putin on hosting the soccer World Cup, which ended on Sunday, saying: "It was beautifully done, so congratulations."
The Trump-Putin summit started with a one-on-one meeting that's expected to last one and a half hours. Later, they will meet with the participation of US and Russian officials.
The meeting is overshadowed by US indictment of 12 Russians last week, who were accused of meddling in the 2016 US Presidential elections.
Trump came under great pressure to raise the alleged meddling with Putin, who has on numerous occasions denied any wrongdoing.
Despite being criticized within the US for being friendly towards Russia, the Trump administration has slapped more sanctions on Moscow.
The two countries are also deadlocked on Russia's role in Syria and Ukraine as well as the Iran nuclear deal, which Washington has torn apart despite continued support from the rest of the parties, including Russia and the European Union.
Possible extension of Strategic Arms Reduction Treaty, known as START, between the two is on the agenda. Trump said: "We have 90 percent of the nuclear, and that's not a good thing, it's a bad thing."
The Intermediate-Range Nuclear Forces Treaty is also expected to be talked about.
European leaders are watching with great concerns the moves of Trump, who attended the two-day North Atlantic Treaty Organisation (NATO) summit in Brussels and then visited Britain last week, and during which Trump publicly battled his traditional allies.