Rakuten president Hiroshi Mikitani has demanded a full explanation from the club and wants to know if they're planning on taking any action against the two players.
Dembele and Griezmann both apologised on Monday but denied there was any racist intent behind their comments, which occurred during the club's tour of Japan in 2019.
"As a sponsor of the club and the organiser of the tour, I regret the players' racist remarks," Mikitani posted on social media on Tuesday.
"Rakuten endorsed Barcelona's philosophy and has supported the club as a sponsor, therefore such behaviour is unacceptable under any circumstances. I will make a formal protest to the club and seek their views."
ESPN has reached out to Barcelona for comment.
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.
"It all took place in Japan, but it could have taken place anywhere on the planet and I would have used the same expression," Dembele wrote on Monday.
"I was therefore not targeting 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 have hurt the people present in these images. Because of this, I would like to sincerely apologise to them."
Griezmann later posted that he was "completely against all forms of discrimination" and that he "refuted 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.
The Japanese e-commerce company extended its link with the club by an additional year last November, until the summer of 2022.
However, due to the coronavirus pandemic, Barca were forced to accept reduced terms of between €30m and €40m, sources told ESPN at the time.
Mikitani is close friends with Gerard Pique and it was his relationship with the Barca defender that led to the formalisation of a sponsorship agreement with the Catalan club.