The video, which was recorded by Dembele, circulated online over the weekend and shows both players laughing at a group of technicians working to fix something in a hotel room.
The footage has been criticised on social media and the France teammates both released statements on Monday to apologise for their behaviour.
"Hello all, during these last few days, a private video dating back to 2019 has been circulating on social media," Dembele wrote on Instagram. "It all took place in Japan. It could've taken place anywhere on the planet and I would've used the same expression.
"I was therefore not targetting any community. I just happened to use this type of expression with my friends regardless of their origins.
"This video has since been made public. I appreciate therefore that it could've hurt the people present in these images. Because of this, I would like to sincerely apologise to them."
"For the past few days, certain people have wanted to paint me as someone that I am not. I firmly refute the accusations made against me and I'm sorry if I have offended any of my Japanese friends."
There has been no confirmation from Barca, the Spanish FA or the France Football Federation about potential punishment for the players.
Both featured for France during Euro 2020 but Dembele's tournament was cut short due to injury, while Griezmann played as Didier Deschamps' side were knocked out on penalties by Switzerland in the round of 16.