Bombalurina, a flirtatious female cat, is best known for singing the song "Macavity" (as a duo on Broadway and as a solo in the West End version).
Corden, meanwhile, will play Bustopher Jones, a smart, well groomed cat with a song named after him.
The announcement means the veil has been lifted on two of the main roles that remained under wraps. Jennifer Hudson had previously been revealed to be playing Grizabella, which the studios have now confirmed.
Rebel Wilson joined the cast last month as Jennyanydots, while Idris Elba was previously announced as Macavity, the "Napoleon of crime" who always escapes after committing a misdeed.
Sir Ian McKellen will star as Gus the Theatre Cat while Dame Judi Dench will play Old Deuteronomy.
Jason Derulo is set to appear as Rum Tum Tugger, while ballerina Francesca Hayward will feature as Victoria, also known as the White Cat.
Filming has begun for the adaptation, which is set to be released on 20 December, 2019.