In the video, recorded during the club's tour of Japan in the summer of 2019, Dembele appears to mock the "ugly faces" of Asian staff at a hotel, with teammate Griezmann laughing.
Both said sorry on social media earlier this week, acknowledging they had been disrespectful but refuting that there had been any racist intent.
"Barcelona deeply regret the displeasure amongst Japanese and Asian fans and partners of our club caused by a video in which two players displayed a lack of respect in their attitude towards several employees in the hotel in which they were staying," a statement read from the Catalan club on Wednesday.
"This attitude in no way coincides with the values that Barcelona represents and defends. The club is committed to improving its education on questions of race, discrimination and diversity. At Barcelona, there is no place for racism or discrimination.
"Barcelona would like to apologise publicly to all the club's fans and partners who feel unhappy about this event from the summer of 2019, a time when the club's responsibilities fell to a board of directors and executive team previous to the current.
"The players have already shown their regret and have apologised to Japanese fans and partners, something that the club values. Nevertheless, Barcelona reserve the right to take the internal measures that it considers appropriate."
The video first emerged on social media over the weekend and the CEO of Rakuten, the club's main sponsor, was the first to express dissatisfaction at the footage.
"Such behaviour is unacceptable under any circumstances," Hiroshi Mikitani posted on Twitter on Tuesday. "I will make a formal protest to the club and seek their views."
A source subsequently confirmed that Griezmann had said sorry to Mikitani personally having already posted an apology on social media.
The France forward wrote that he was "completely against all forms of discrimination" and he "refuted the accusations made against me and I'm sorry if I have offended any of my Japanese friends."
The fallout continued on Wednesday when Konami, who are also one of Barca's partners in Asia, announced they had cancelled Griezmann's contract to work as an ambassador for the entertainment company.
Dembele's apology said: "It all took place in Japan, but it could have taken place anywhere on the planet and I would have used the same expression.
"I was therefore not targeting any community. I just happened to use this type of expression with my friends regardless of their origins.
"I appreciate that it could have hurt the people present in these images. Because of this, I would like to sincerely apologise to them."