Trump says he wanted to share with Russian Federation "facts pertaining to terrorism and airline flight safety".
Trump went on to tweet that, as president, he had the "absolute right" to declassify and share any such information related to terrorism with Russian Federation.
"It's hard to imagine what else can these people who generate such nonsense and rubbish can dream up next", said Putin.
The Washington Post reported Monday that Trump divulged highly classified "code-word" information that could enable the Russians to trace the source of the intelligence.
The Post reported that Trump shared classified information with Russian Federation during a meeting last week at the Oval Office, which national security experts found alarming.
When questioned by the press, McMaster did not answer a question about whether the information shared by Trump was secret or not.
That appeared to put him at odds with members of his own senior staff. Secretary of State Rex Tillerson and National Security Adviser H.R. McMaster denied on Monday that Trump had revealed intelligence sources and methods or military operations at the Russian Federation meeting. His action raised fresh questions about his handling of classified information and his dealings with Russian Federation, which is widely considered an adversary by many US officials and Western allies.
Speaking at a joint news conference with the visiting Italian prime minister, Putin said he would be willing to turn over notes of Trump's meeting with the Russian diplomats if the White House agreed.
The president can also declassify or share any information he wants.
US Defense Secretary Jim Mattis told reporters Tuesday that the uproar surrounding the Trump administration's handling of confidential information "never even came up" in his most recent round of talks with US allies.
Earlier Tuesday, Trump took to Twitter to defend his reported decision to give classified intelligence information to Russian Federation, in a statement that appeared to contradict earlier denials by White House officials, who had said he had not let slip any secrets. Trump and his associates have said that any disclosures made were appropriate in the context of the conversation and justified and under his authority.
After Trump disclosed the information, which one of the officials described as spontaneous, officials immediately called the CIA and the National Security Agency, both of which have agreements with a number of allied intelligence services, and informed them what had happened.
In his tweets, Trump did not dispute media reports, initially from The Washington Post, that he revealed details that could jeopardize intelligence capabilities. Associated Press writer Paisley Dodds contributed from London.
And the way in which Trump's tweets contradicted his administration's initial explanation for echoed Trump's firing of Comey.