He also said they "will probably get together in the not-too-distant future" to talk about the global arms race and other mutual concerns, such as the war in Syria and North Korea's development of nuclear weapons.
Trump added the arms race between the USA and Russian Federation "is getting out of control. but we will never allow anybody to have anything close to what we have".
"I had a call with President Putin and congratulated him on the victory, his electoral victory", Trump said on Tuesday. Before the election even began, the impartiality of the results was compromised.
The White House press office declined to comment on the briefing materials given to Trump.
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. "As you know, he made a statement that being in an arms race is not a great thing - that was right after the election, one of the first statements he made".
Trump's decision Tuesday to ignore his advisers' guidance and congratulate Putin adds another layer to an emerging portrait in which the U.S. president has demonstrated a reluctance to confront the Russian leader amid an ongoing investigation into whether the Trump campaign colluded with Moscow to tilt the 2016 election in his favor.
They also discussed the crises in Ukraine and Syria, stressing the importance of securing progress as soon as possible, the Kremlin said.
The Kremlin also said Trump had congratulated Putin. President Trump also floated the idea of working with Russian Federation in the near future.
Arizona Senator John McCain issued a sharp rebuke of Donald Trump for his call to Vladimir Putin, and Nancy Pelosi noted that Trump and his national security team have still not taken steps to protect the upcoming midterm elections from more Russian meddling.
According to a White House statement, "President Trump congratulated President Putin on his March 18 re-election, and emphasized the importance of denuclearizing the Korean Peninsula".
"It was agreed to develop further bilateral contacts in light of the changes in leadership at the US Department of State".
The call had "a focus on overcoming the accumulated problems in Russian-American relations", the Kremlin said.
"The leaders spoke in favour of developing practical cooperation in different spheres, including on questions of how to ensure strategic stability and fight global terrorism", the Kremlin said in a statement.
Putin has struck a softer tone toward the West since Sunday, saying he has no desire for an arms race and would do everything he could to resolve differences with other countries.
"Did poisoning in the United Kingdom come up in the call?" a reporter asked.