We investigate whether Higgs inflation can occur in the Standard Model starting from natural initial conditions or not. The Higgs has a non-minimal coupling to the Ricci scalar. We confine our attention to the regime where quantum Einstein gravity effects are small in order to have results that are independent of the ultraviolet completion of gravity. At the classical level we find no tuning is required to have a successful Higgs inflation, provided the initial homogeneity condition is satisfied. On the other hand, at the quantum level we obtain that the renormalization for large non-minimal coupling requires an additional degree of freedom that transforms Higgs inflation into Starobinsky $R^2$ inflation, unless a tuning of the initial values of the running parameters is made.