A singular perturbation method is employed in order to develop an analytical solution to the problem of the unsteady mass transfer from a sphere immersed in an unbounded saturated porous medium. Al the inception of the process. the sphere is suddenly leaking a contaminant, which spreads in the porous medium by convection and diffusion. The boundary conditions at the surface of the sphere are either constant concentration or constant mass flux. Throughout the process the Peclet number is small but finite. The time and length domains of the problem are separated in four subdomains, which result from the combinations of short and long times, and inner and outer regions. Based on the physical analysis of the problem, the governing equations in these regions are derived and solved in the time domain or the Laplace domain. A matching technique is used to derive the final expressions for the contaminant concentration field and the mass transfer coefficients. Hence, analytical asymptotic solutions for the concentration of the contaminant in the entire space and time domains are derived in terms of the Peclet numbers. The solutions are validated by comparison with known analytical results.