He is heard seeming to say "It's green only, n**" during a video which seems to show him and bandmate Zayn Malik sharing a joint.
The three-letter word was taken by some to be an abbreviation of the n-word.
The footage, published by the Mail Online website, was apparently filmed by Tomlinson during their tour in Peru.
At one point smoke is seen being blown across the car.
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."
Some fans expressed their disappointment at the alleged language used.
One wrote on Twitter: "Worst part about the video was hearing Louis say 'n**'."
The pair also faced criticism from John Whittingdale, chairman of the Commons culture, media and sport select committee.
The Tory MP told the Daily Mail: "They are adored by many people and they have a responsibility to set an example."
A spokesman for the band previously said: "This matter is in the hands of our lawyers."
He said today there was no further statement.
The group, who have a huge following on social media, rose to fame as teenagers on ITV's The X Factor talent show and have become one of Britain's biggest bands, with a huge and loyal following of young girls across the world.
Other band members Harry Styles, Liam Payne, and Niall Horan are believed to have been travelling in different cars and were not seen in the video.
Earlier this month, the band were named fourth in the Sunday Times Rich List of young musicians with an estimated fortune of £14 million each and a combined wealth of £70 million.
PR consultant Mark Borkowski said the controversy is likely to have been a "shock" to the band's management.
He said: "It changes the focus. I guess they are no longer the squeaky-clean boys."
Discussing how the incident would affect One Direction's ability to attract sponsorship deals, he said: "It definitely will affect certain things. Brands do not want to be associated with the negative headlines that we are reading this morning."
Ahead of the opening of the UK leg of their Where We Are tour in Sunderland last night, fans who queued in heavy rain were keen to show their support for the band despite the cannabis allegation.
Outside the soaking Stadium of Light, where 52,000 screaming fans watched the show, Shelby Clark, 20, from Newcastle, said: "They could be doing worse, it's not that bad."
Her friend, Samantha Stirling, also 20 and from Newcastle, said: "At the end of the day, they are just young lads.
"I know it's prohibited but they are under so much stress.
"It doesn't put us off - we wouldn't be here if it did."