Background/Objectives: Radial artery tracking (RAT) in the short-axis view is a pivotal step for ultrasound-guided radial artery catheterization (RAC), which is widely employed in various clinical settings. To eliminate disparities and lay the foundations for automated procedures, a pilot study was conducted to explore the feasibility of U-Net and its variants in automatic RAT. Methods: Approved by the institutional ethics committee, patients as potential RAC candidates were enrolled, and the radial arteries were continuously scanned by B-mode ultrasonography. All acquired videos were processed into standardized images, and randomly divided into training, validation, and test sets in an 8:1:1 ratio. Deep learning models, including U-Net and its variants, such as Attention U-Net, UNet++, Res-UNet, TransUNet, and UNeXt, were utilized for automatic RAT. The performance of the deep learning architectures was assessed using loss functions, dice similarity coefficient (DSC), and Jaccard similarity coefficient (JSC). Performance differences were analyzed using the Kruskal–Wallis test. Results: The independent datasets comprised 7233 images extracted from 178 videos of 135 patients (53.3% women; mean age: 41.6 years). Consistent convergence of loss functions between the training and validation sets was achieved for all models except Attention U-Net. Res-UNet emerged as the optimal architecture in terms of DSC and JSC (93.14% and 87.93%), indicating a significant improvement compared to U-Net (91.79% vs. 86.19%, p < 0.05) and Attention U-Net (91.20% vs. 85.02%, p < 0.05). Conclusions: This pilot study validates the feasibility of U-Net and its variants in automatic RAT, highlighting the predominant performance of Res-UNet among the evaluated architectures.