Bamba was offered the job on Tuesday and has been given the blessing of the outgoing Sabri Lamouchi, whom he was assistant to.
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Despite strong links with Nathan Jones, it is the Ivorian who will become the next Bluebirds' boss - their third this calendar year.
Bamba announced his retirement from football at the end of last season but quickly made a return to the club he represented for five years, as part of Lamouchi's backroom team.
Lamouchi announced his departure as Cardiff boss after guiding the club to Championship safety.
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It was a tense battle for survival after spending much of the season staring relegation in the face.
However, after a meeting with owner Vincent Tan last week, the decision was made that Lamouchi would not continue in the post after initially being appointed in January on a short-term deal.
Lamouchi was a fan-favourite at the club and took to social media to thank Bluebirds fans for their support during his tenure.
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"As my journey as coach at @cardiffcityfc comes to a close, I would like to thank everyone at the club," Lamouchi wrote on Instagram.
"It has been an honour to be entrusted with this brief mission, and to have successfully fulfilled it. The players fought their best to keep the club at this level, and I hope this will be a starting point for next season.
"While our paths may diverge, I feel grateful for the opportunity that was given to me, and wish everyone the very best in the future. To the fans, thank you for your support everywhere we played, it meant a lot to me.
"Sabri."