A theoretical analysis of convective instability driven by buoyancy forces under the transient concentration fields is conducted in an initially quiescent, liquid-saturated, and anisotropic cylindrical porous layer supported by a gas phase. Darcy's law and Boussinesq approximation are used to explain the characteristics of fluid motion, and linear stability theory is employed to predict the onset of buoyancy-driven motion. Under the quasi-steady-state approximation, the stability equations are derived in a similar boundary layer coordinate and solved by the numerical shooting method. The critical Ra-D is determined as a function of the anisotropy ratio. Also, the onset time and corresponding wavelength are obtained for the various anisotropic ratios. The onset time becomes smaller with increasing Ra-D and follows the asymptotic relation derived in the infinite horizontal porous layer. Anisotropy effect makes the system more stable by suppressing the vertical velocity.