The pair recently reunited their band, The Libertines - putting their differences, which originally caused the group's end - behind them, and Carl is keen for Pete to provide vocals on the track 'So Long, My Lover' on his self-titled album, which will be released in October.
Carl said: "It's really brutally honest and a truthful telling of a time - I hope people can relate to it."
The album tells the story of a break-up, but Carl refuses to say whether it is about his split from long-term partner Annalisa Astarita or his fall-out with Pete, who was thrown out of The Libertines for his out-of-control drug addiction.
Carl added to The Sun newspaper: "I'm a happier and more complete person for having made this record. This album is the truth."
32-year-old Carl will play live with the reunited line up of The Libertines for the first time in six years at the Reading and Leeds festivals in the UK later this month. It is unlikely he will debut any of his solo tracks live however, as they mark a significant departure from the sound he has pursued in other bands.
A spokesperson for the rocker said: "Having released two albums with The Libertines and then a further two with Dirty Pretty Things, Carl was very keen to do his own record and do things his own way.
"It is a departure from Carl's previous releases, but maintains his ear for melody. The album allowed him to work with new people and try new musical ideas. The result is a very honest and heartfelt record, with upbeat tales of friendship and laments for lost love."