Speaking at a press conference, Putin said the transcript could be provided "if the US administration finds it appropriate".
The Trump administration has come under fire over its handling last week of the firing of FBI Director James Comey and then revelations earlier this week that the President provided highly-classified intelligence information to Russian officials, CNN reported.
Trump met Lavrov and Russian Ambassador Sergey Kislyak at the White House on May 10.
According to a report in the Washington Post, Trump revealed to them valuable intelligence information on the Islamic State (IS) militant group.
The US President on Tuesday hit back at the accusations, claiming it is his right to share information on security as he sees fit.
But he did not confirm or deny whether the information was classified.