The 35-year-old has started chemotherapy and will be supported by the Bluebirds' medical team.
A club statement said: "Universally admired by teammates, staff and supporters in the Welsh capital, Sol has begun his battle in typically positive spirits and will continue to be an integral part of the Bluebirds family.
"All future updates on Sol's progress will be provided only via official Club channels.
"While we request privacy for him and his family at this time, messages of support to be passed on to Sol may be sent to [email protected]?
"We are all with you, Sol."
Bamba has made 117 appearances for Cardiff, scoring 10 goals in the process, since joining them from Leeds in October 2016, with his most recent game for the Bluebirds coming in the 3-2 home victory over Birmingham on December 16.
He helped the club into the Premier League in 2018 and has also had spells at Hibernian and Leicester after starting out at Paris Saint-Germain.
Bamba - born in Paris but to Ivorian parents - chose to play for the Ivory Coast and represented them at the 2008 Olympics and was also part of the side that reached the 2012 Africa Cup of Nations final.