Elastography Ultrasound provides elasticity information of the tissues, which is crucial for understanding the density and texture, allowing for the diagnosis of different medical conditions such as fibrosis and cancer. In the current medical imaging scenario, elastograms for B-mode Ultrasound are restricted to well-equipped hospitals, making the modality unavailable for pocket ultrasound. To highlight the recent progress in elastogram synthesis, this article performs a critical review of generative adversarial network (GAN) methodology for elastogram generation from B-mode Ultrasound images. Along with a brief overview of cutting-edge medical image synthesis, the article highlights the contribution of the GAN framework in light of its impact and thoroughly analyzes the results to validate whether the existing challenges have been effectively addressed. Specifically, This article highlights that GANs can successfully generate accurate elastograms for deep-seated breast tumors (without having artifacts) and improve diagnostic effectiveness for pocket US. Furthermore, the results of the GAN framework are thoroughly analyzed by considering the quantitative metrics, visual evaluations, and cancer diagnostic accuracy. Finally, essential unaddressed challenges that lie at the intersection of elastography and GANs are presented, and a few future directions are shared for the elastogram synthesis research.