Trump's overture to Putin was criticized by top senators in his own Republican Party, who called Sunday's election a sham.
They spoke after Washington slapped a new round of sanctions against Moscow created to punish Russians responsible for attempts to interfere in the 2016 presidential elections.
President Trump meets with Russian President Vladimir Putin at the G-20 Summit in Hamburg, Germany on July 7, 2017. The Skripals remain in a critical condition.
The Kremlin said the conversation between the usa and Russian presidents was "constructive and business-like", adding that Russia hoped to "overcome problems" that had arisen between the two nations.
Crises in Syria and Ukraine were also discussed.
Trump did tell reporters that he would meet with Putin in the "not too distant future". Two people familiar with the note cards acknowledged that they included instructions not to congratulate Putin.
Another unnamed senior White House official told the outlet, however, National Security Adviser H.R. McMaster did not mention the issue during a telephone briefing with the president.
Trump's comments drew significant criticism among USA politicians, including by Arizona GOP Sen. Many others, including German Chancellor Angela Merkel, sent congratulatory telegrams. "I wouldn't read much into it", Corker said.
Putin won re-election with 76.6% of the vote.
Apparently not discussed: Russian efforts to interfere in the 2016 US presidential campaign and the poisoning of former Russian spies in the United Kingdom with a Russian-made nerve agent.
President Trump's eagerness to meet with Russian Federation in the near future comes at a tense time between the two nations.
Tensions between the Moscow and the West continues to mount amid worldwide concern over a nerve gas attack on a former Russian double agent living in England and questions about meddling in elections and attempting to hack infrastructure in the U.S.
Putin denied that Russian Federation had any role and called the claim "nonsense".
President Obama called the Kremlin after Putin's election in 2012, touting the success of a "reset" in relations under former Russian President Dmitry Medvedev. Trump has not been highly critical of Russia in public, while maintaining there was no collusion between his presidential campaign and Russians. "We've placed tough sanctions on Russian Federation and a number of other things where we have shown exactly what our position is".
In addition, Trump and Putin expressed satisfaction with the apparent easing of tensions over North Korea's weapons programme, according to the Kremlin.
Nauert is citing a preliminary report by the Organization for Security and Cooperation in Europe that says Russia's election took place in an overly controlled environment that lacked an even playing field for all contenders.
White House Press Secretary Sarah Huckabee Sanders confirmed that Trump spoke with Putin on Tuesday morning. "What we do know is that Putin has been elected in their country, and that's not something that we can dictate to them how they operate".