Assuming a simple form for the growth index γ(z) depending on two parameters γ0≡γ(z=0) and γ1≡γ′(z=0), we show that these parameters can be constrained using background expansion data. We explore systematically the preferred region in this parameter space. Inside general relativity we obtain that models with a quasistatic growth index and γ1≈-0.02 are favored. We find further the lower bounds γ00.53 and γ1-0.15 for models inside GR. Models outside GR having the same background expansion as ΛCDM and arbitrary γ(z) with γ0=γ0ΛCDM, satisfy Geff,0>G for γ1>γ1ΛCDM, and Geff,0<G for γ1<γ1ΛCDM. The first models will cross downwards the value Geff=G on very low redshifts z<0.3, while the second models will cross upwards Geff=G in the same redshift range. This makes the realization of such modified gravity models even more problematic.