  We study the cosmological evolution of some nonlocal gravity models, when the
initial conditions are set during a phase of primordial inflation. We examine
in particular three models, the so-called RT, RR and $\Delta_4$ models,
previously introduced by our group. We find that the RR and $\Delta_4$ models
have a stable evolution also during inflation. The RT model has an apparent
instability, but we show that, because of the smallness of the scale associated
to the nonlocal term compared to the inflationary scale, this instability is
innocuous and also the RT model has a viable evolution even when its initial
conditions are set during a phase of primordial inflation.
