The video, obtained by the Daily Mail newspaper, appears to have been filmed by 22-year-old Tomlinson in the back of a car in Peru during the South America leg of the boyband's world tour.
Tomlinson says: "So here we are, leaving Peru. Joint lit. Happy days!"
At one point smoke is seen being blown from behind the camera.
A motorcycle police officer drives past and Tomlinson says: "He's having a look. He's thinking 'I'm sure I can smell an illegal substance in there'. And he's hit the nail on the head."
Tomlinson and Malik are also heard joking about "illegal substances" as they smoke in the back of a vehicle.
The band's other members -- frontman Harry Styles, Liam Payne, and Niall Horan -- are believed to have been travelling in different cars and are not seen in the video.
Possession of a small amount of cannabis is not illegal in Peru.
A spokesman for the band, who start a run of shows in Britain on Wednesday, said: "This matter is in the hands of our lawyers."
Thousands of fans, or "Directioners", have taken to Twitter to defend the band, while others have said they destroyed or sold their tickets for upcoming concerts.
Put together for a British television singing contest in 2010, they have developed a giant, fiercely loyal following of young girls worldwide.
Their third studio album, "Midnight Memories", raced to the top of the US Billboard Top 200 chart, making the five-piece the first group in US chart history to debut at number one with their first three albums.
Their video for "Best Song Ever" has attracted nearly 250 million views on YouTube.
The Sunday Times newspaper's Rich List 2014, out earlier this month, reckoned the bandmembers were worth £14 million ($23 million, 17 million euros) each.
source: www.abs-cbnnews.com
