Visible-infrared person re-identification seeks to retrieve images of the same individual captured over a distributed network of RGB and IR sensors. Several V-I ReID approaches directly integrate both V and I modalities to discriminate persons within a shared representation space. However, given the significant gap in data distributions between V and I modalities, cross-modal V-I ReID remains challenging. Some recent approaches improve generalization by leveraging intermediate spaces that can bridge V and I modalities, yet effective methods are required to select or generate data for such informative domains. In this paper, the Adaptive Generation of Privileged Intermediate Information training approach is introduced to adapt and generate a virtual domain that bridges discriminant information between the V and I modalities. The key motivation behind AGPI^2 is to enhance the training of a deep V-I ReID backbone by generating privileged images that provide additional information. These privileged images capture shared discriminative features that are not easily accessible within the original V or I modalities alone. Towards this goal, a non-linear generative module is trained with an adversarial objective, translating V images into intermediate spaces with a smaller domain shift w.r.t. the I domain. Meanwhile, the embedding module within AGPI^2 aims to produce similar features for both V and generated images, encouraging the extraction of features that are common to all modalities. In addition to these contributions, AGPI^2 employs adversarial objectives for adapting the intermediate images, which play a crucial role in creating a non-modality-specific space to address the large domain shifts between V and I domains. Experimental results conducted on challenging V-I ReID datasets indicate that AGPI^2 increases matching accuracy without extra computational resources during inference.