White House chief of staff John Kelly is said to be "furious" over the leaks to the Post, which broke the news of Trump ignoring notes about the call on Tuesday evening, and plans to address the matter with aides on Wednesday in an attempt to figure out who leaked the warning.
The Washington Post reported that Trump offered his congratulations despite a written briefing by national security advisers urging him not to do so.
Ukrainian President Petro Poroshenko did not congratulate Putin for his election win, but instead released a statement that criticized Russian Federation for holding the vote in Crimea, territory that was annexed from Ukraine in 2014.
In calling Putin, Trump was following precedent.
"The President calls Putin - not to condemn him for using a nerve agent in Britain or interfering in our election - but to congratulate him on winning an election in which his top opponent was prohibited from running", Schiff, of California, wrote on Twitter.
Trump's congratulating Putin has revived criticism that he has been too reluctant to criticize the Russian leader and retaliate for Russian meddling in the 2016 presidential election that Trump won.
But Trump's defenders were quick to point out that his predecessor, President Barack Obama, had also congratulated Putin after his successful presidential run.
Republicans in Washington were critical of Trump's decision to congratulate Putin, but also lined up behind the president in anger about the leak.
Sen. Jeff Flake, a noted Republican critic of Trump, disputed Sanders' remarks.
The White House statement earlier Wednesday about a possible firing was an unusual threat and an indication of the seriousness with which the administration is treating the latest breach. "Leaking such information is a fireable offense and likely illegal", said a senior White House official, who requested anonymity. "I'm always reminded of the election they used to have in nearly every communist country, where whoever the dictator was at the moment always got a huge percentage of the vote".
I am unconvinced by Clara Holzer's claim that US President Donald Trump is "correct to say that we should wait for the facts before commenting" on the United Kingdom spy poisoning. Russian Federation has denied the accusation.
Putin received calls from a number of other foreign leaders, including French President Emmanuel Macron, Chinese President Xi Jinping, Japanese Prime Minister Shinzo Abe and Indian Prime Minister Narendra Modi.
Last month, special counsel Robert Mueller indicted 13 Russian individuals and three organizations on charges of interfering in the election.
Perhaps more important than the call itself is the fact that the contents of the president's briefing materials were leaked to the national press.