The Kremlin leader describes the footage as showing Russian forces bombing "international bandits" in Syria and comments on the manoeuvres in detail.
But Russian-language news site Meduza said the video was filmed in 2009 and actually shows the US bombing the Taliban in Afghanistan.
It said the footage was posted on US military and veteran website Military.com.
Putin's spokesman Dmitry Peskov denied reports the video was misidentified, telling journalists: "I can confirm that this is indeed the defence ministry's material. The defence minister gave this to the president."
He said reports alleging otherwise were "incorrect and inappropriate."
Russian investigative site Conflict Intelligence Team posted a video comparing the US video with the one demonstrated by Putin, which appear to be identical.
Journalists speculated that Putin had been handed not defence ministry footage but a Russian-language YouTube video that has been viewed more than 75,000 times.
The video identifies the footage as in Syria and gives a Russian voiceover. But the video in fact shows Apache helicopters in Afghanistan, a military expert Andrei Payusov told RBK news site.
Putin, who is the supreme commander of the armed forces, has rarely been shown in public using a smartphone, unlike Prime Minister Dmitry Medvedev who is a fan of Apple devices.
Peskov said the phone belonged to a member of staff but refused to be more specific. Putin is shown handing the phone back to a dark-suited man.
The footage comes in the third episode of three-times Oscar-winning Stone's documentary series on Putin, which is being aired on US cable network Showtime and Russia's Channel One state television.
Even some pro-Kremlin media has covered the reports and posted comparison videos.
Leonid Volkov, campaign chief of jailed opposition leader Alexei Navalny mocked the report as showing Putin was being misinformed by advisors.
"Putin also often gets shown the exchange rate of the bitcoin and is told it's the reserve fund growing," Volkov wrote on Twitter.
(This story has not been edited by NDTV staff and is auto-generated from a syndicated feed.)