We carry out a linear stability analysis for a free boundary between two fluids in a Hele-Shaw cell, which is driven by the Darcy's law and an injection or suction at the center. In general, stability of an interface is determined by the well-known Saffman-Taylor instability condition. The linear growth rate of the perturbation depends on two parameters; the rate of the injection/suction and the viscosity contrast of the two fluids. In this paper, we numerically find a parameter region, in which the interface can be stable (resp. unstable) even though it has been considered to be unstable (resp. stable) due to the Saffman-Taylor instability. For the case of the injection, it is suggested that such parameter region vanishes as time increases to infinity. However, the destabilization can be retarded for a sufficiently long time, as one tunes the viscosity and the injection rate of the injecting fluid.