Materials exhibiting hyperbolic dispersion have attracted considerable interest within the optical and quantum communities due to their potential to confine electromagnetic waves at subwavelength scales, thereby surpassing the diffraction limit. This capability arises from the support of high‐k (i.e., wavevector) eigenmodes in hyperbolic materials. However, a substantial trade‐off exists between energy confinement and dissipation of these high‐k eigenmodes under ambient conditions, with the fundamental limits of this trade‐off remaining unexplored. Herein, the concept of hyperbolic asymptotic eigenmodes is studied, which are high‐k eigenmodes that align with the asymptotic direction, as the name implies. Additionally, the challenge of mitigating the losses associated with these modes in hyperbolic metamaterials is addressed. This is achieved by carefully tuning the permittivity tensors of these metamaterials to satisfy loss compensation, where the ratio of the real to imaginary parts of the perpendicular and parallel components is equal. The findings demonstrate that hyperbolic asymptotic eigenmodes exhibit strong confinement and long‐range propagation. Furthermore, proof‐of‐concept simulations for hyperbolic asymptotic eigenmodes in active hyperbolic metamaterials with gain molecules at visible frequencies are presented. The proposed loss compensation of hyperbolic asymptotic eigenmodes holds promise for advancing both conventional and quantum electromagnetic signal processing.