US President Donald Trump has spoken to his Russian counterpart Vladimir Putin to congratulate him on his electoral victory two days ago.
He said they would meet in the "not too distant future" to discuss limiting a growing arms race, Ukraine and Syria.
Mr Putin was re-elected by a landslide, with more than 76% of the vote, for a fourth six-year term.
But there was no strong challenger, with the main opposition leader, Alexei Navalny, barred from the race.
Mr Trump said the arms race between the US and Russia was "getting out of control... but we will never allow anybody to have anything close to what we have".
According to the Washington Post, citing officials it said were familiar with the matter, the call was made despite warnings from Mr Trump's national security advisers, who provided a briefing which included a section that read "DO NOT CONGRATULATE".
During the call, Mr Trump did not mention the issue that has sparked growing Western tensions with Moscow - the poisoning of ex-spy Sergei Skripal and his daughter Yulia in Britain.
The UK government blamed the nerve agent attack on Russia, which denies any involvement.
The Kremlin said the conversation between the US and Russian presidents was "constructive and business-like", adding that Russia hoped to "overcome problems" that had arisen between the two nations.
"An American president does not lead the free world by congratulating dictators on winning sham elections. And by doing so with Vladimir Putin, President Trump insulted every Russian citizen who was denied the right to vote in a free and fair election to determine their country's future, including the countless Russian patriots who have risked so much to protest and resist Putin's regime," McCain said in a statement.
EU Commission chief Jean-Claude Juncker also wrote a letter of congratulations to the Russian leader on Tuesday, pledging to "always be a partner" in improving security co-operation with the Kremlin.
"I have always argued that positive relations between the European Union and Russian Federation are crucial to security of our continent," he said in the letter, which he shared on Twitter.
But the leader of the UK Conservative Party's group in the European Parliament, MEP Ashley Fox, said Mr Juncker's letter was "disgraceful" and "nauseating" in its "failure to mention Russia's responsibility" in the attack on Mr Skripal and his daughter.
German Chancellor Angela Merkel was among the first Western leaders to "warmly congratulate" Mr Putin on Monday - a day after his re-election - stressing the need to continue dialogue "to address important bilateral and international challenges and find viable solutions".
French President Emmanuel Macron wished Mr Putin success in "modernising Russia", but urged Moscow to shed light on the "unacceptable" attempted assassination of an ex-spy in Britain.
On Tuesday, 23 Russian diplomats expelled by the UK left London.
Moscow has also ordered 23 British diplomats to leave in a tit-for-tat response.
The US recently imposed sanctions on a group of Russians over alleged interference in favour of the election of Donald Trump as US president in 2016.