Trump said he was not aware of the general's comments, adding that no evidence or intelligence of the alleged bounty ever 'reached my desk'.
Trump said he had "never discussed" the story with Putin.
Trump spoke on the phone with Putin on July 23, the eighth conversation the two leaders have had since the intelligence was reportedly included in the president's written daily intelligence briefing in late February.
When pressed on intelligence reports that Russia has supplied weapons to the Taliban, Trump seemed to justify the Russians' actions by citing the fact that the US sold weapons to the Taliban when the group was fighting the Soviet Union decades ago.
"No, that was a phone call to discuss other things", Trump told Axios on HBO.
POTUS has faced a harsh wave of criticism from his opponents over his lack of a reaction to unconfirmed reports in the media about alleged bounties that the Kremlin purportedly paid to the Taliban for the killing of American troops in Afghanistan.
US Secretary of State Mike Pompeo said Thursday that the deaths of "300 Russians" in Syria sent a warning to Moscow, as he defended the administration as tough on President Vladimir Putin.
"I would, I have no problem with it", Trump said.
President Donald Trump's niece Mary Trump is continuing to talk to reporters about her new tell-all book, which has become a runaway success, MSNBC's Joy Reid described.
"The idea that somehow he didn't know or isn't being briefed, it is a dereliction of duty", Biden said.
Russian Federation has denied the allegations, and the White House insisted that there is "no consensus" that the intelligence is accurate.
The issue has also been made into an election issue, with presumed Democratic nominee and other Democrats taking aim at the president for his handling of the issue.
National Security Adviser Robert O'Brien has said Trump was not verbally briefed on the intelligence because his CIA briefer concluded the reports were uncorroborated.