European security official speaking with the AP said the move could have an effect on the trust between the US and intelligence sharing partners.
Secretary of State Rex Tillerson also disputed the Post report.
The Wednesday meeting between Trump, Lavrov and Kislyak had already raised alarm bells in Washington, primarily because it came one day after Trump chose to fire Comey while the bureau investigated his campaign's alleged ties to Russian Federation.
National Security Adviser H.R. McMaster said the president's comments were "wholly appropriate".
In January, an Israeli newspaper reported that American intelligence officials warned their Israeli counterparts not to share sensitive information with the Trump administration because of the threat that it could be leaked to Russian Federation.
"During that exchange the nature of specific threats were discussed, but they did not discuss sources, methods or military operations".
The official said Tuesday that sharing the information "could be a risk for our sources".
Some of the leaders Trump will meet come from countries the USA has intelligence-sharing agreements with.
The U.S. and Western officials all spoke on the condition of anonymity in order to discuss sensitive information.
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Trump's tweets did not address whether he revealed classified information about the Islamic State group, as published reports have said and as a US official told The Associated Press on Tuesday.
The information was provided to the U.S.by Israel, NBC News reported.
"While it is fully within the President's powers to manage classified information and direct our country's foreign policy, if classified information was divulged without good reason, it could not only be extremely imprudent, but also hamper our ability to gather intelligence in the future". Asked if he had concerns about the president's handling of classified information, the Republican Senate Majority Leader Mitch McConnell let out a small laugh and before replying simply, "No".
A top German lawmaker, Burkhard Lischka, who sits on the Intelligence Oversight Committee in the German parliament, said that "if it proves to be true that the American president passed on internal intelligence matters that would be highly worrying".
But in an interview with NBC's Lester Holt, Trump later admitted that he had made the decision to fire Comey well before Rosenstein's memo, in part because he was frustrated by the director's investigation into possible collusion between his presidential campaign and the Russian government. He's questioned the competence of intelligence officials, challenged their assessment that Russian Federation meddled in last year's election to help him win, and accused them of leaking information about him and his associates.
The revelation also prompted cries of hypocrisy.
The controversy engulfed the White House.
"Revealing classified information at this level is extremely unsafe and puts at risk the lives of Americans and those who gather intelligence for our country".
The White House vigorously denied it all. Now it's up to Sean Spicer, who will brief the press this afternoon, not to muck up the messaging. AP writer Paisley Dodds contributed from London.
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