The White House confirmed with CNN and NBC News that the meeting took place, the news outlets reported.
During the meeting, Trump spoke alone with the Russian leader, joined only by Putin's translator, an arrangement that Bremmer said was a breach of national security protocol.
The informal meeting came just hours after the two leaders met face-to-face for formal talks that were scheduled to last a half an hour but lasted two hours.
During the July 7 meeting, Putin and Trump discussed numerous issues, including conflicts in Syria and Ukraine, cybersecurity and the fight against terrorism.
The two leaders also agreed to implement a regional ceasefire in southern Syria.
Trump has said that he would invite Putin to the White House at "the right time," adding that a dialogue with the Russian government was necessary.