Morphogenetic theories investigate the creation and the emergence of form in living organisms. A novel approach for studying free boundary problems during morphogenesis is proposed in this work. The presence of mass fluxes inside a biological system is coupled with the local gradient of diffusing morphogens. The contour stability of a growing material is studied using a two-dimensional system model with a rectilinear free border inside a Hele-Shaw cell. Modeling mass transport during morphogenesis allows fixing the velocity at the traveling wave solution as a function of one-dimensionless parameter. Performing a perturbation of the free boundary, the dispersion relation is derived in an implicit form. Although both the velocity of the moving front and the surface tension act as stabilizing effects at small wavelengths, the dispersion diagrams show that the rectilinear border is always unstable at large wavelengths. Further applications of this model can help give insights into a number of free boundary problems in biological systems.