The network will announce the news on Tuesday (November 15), according to Entertainment Weekly.
Early Show already features Erica Hill, Chris Wragge and Jeff Glor, but the show has been struggling in the ratings compared to its early-morning rivals.
In July, former CBS Early Show host Harry Smith left the network after 25 years for rival network NBC.
Rose is the host of his own show on PBS, while King, who is Oprah Winfrey's best friend, has her own programme on Winfrey's network OWN.
King once revealed that she has been offered the chance to participate on ABC's Dancing with the Stars three times but turned it down each time because she "can't dance".
Watch a video of Gayle King interview rapper 50 Cent on her own talkshow below (US only):