We consider the one-dimensional one phase Stefan problems with temperature or thermal flux boundary conditions. Exact and closed-form solutions for these two free boundary problems are very limited and highly restricted to particular initial and boundary conditions, and so are the similarity solutions. In general, only numerical methods allow one to completely solve such nonlinear problems. In this paper, the quasi-similarity solutions are defined and, for each problem, an analytical approximation of the interfacial position is given as function of the time-dependent boundary datum.