The much-anticipated joint album between J. Cole and Kendrick Lamar never materialized, and Cole recently addressed the long-standing rumors during an episode of the A Safe Place podcast, co-hosted by Lil Yachty and MitchGoneMad. Lil Yachty brought up the "huge, infamous, mystery" that emerged over a decade ago when Kendrick Lamar teased a collaboration with Cole.
Cole confirmed that he and Kendrick had indeed discussed the project early in their careers but never had the opportunity to fully pursue it. The idea surfaced before Kendrick's first album and possibly before Cole's as well. They had a conversation about the project when Kendrick visited Cole at producer No ID's studio. Cole had shared some beats, some of which already had hooks or verses, and they talked about doing a joint project. Kendrick even tweeted about it and shared a picture of them, generating excitement among fans.
However, the collaboration didn't come to fruition due to their busy schedules and the time commitment required for a project of that magnitude. Cole explained that creating something of their full potential would take at least a year, and with both artists being successful and having personal lives and families, finding that time was a challenge. Although the rumors about the joint album were put to rest years ago, Cole clarified that there isn't an entire album sitting on a shelf somewhere.
In short, while the idea of a collaborative project between J. Cole and Kendrick Lamar was discussed and even worked on to some extent, it never progressed into a complete album, and there are no unreleased songs sitting in a vault.