In this paper the scaling of the mean velocity profile and Reynolds stresses is considered for the case of turbulent near-wall flows subjected to strong pressure gradients. Strong pressure gradients are defined as those in which the streamwise pressure gradient non-dimensionalized with inner variables, $p_x^+$, is greater than 0.005. A range of values of this parameter (${-}0.02\,{<}\,p_x^+\,{<}\,0.06$) is examined in this paper. An appropriate functional form for the mean velocity profile is developed and used to parameterize available data. A physical model for the parametric variation with pressure gradient is then developed. This model is based on the concept of a universal critical Reynolds number for the sublayer which explains (both qualitatively and quantitatively) the variation of the important parameters in the inner flow. In particular this gives an explanation for the shift in the apparent log-law due to pressure-gradient effects and provides an appropriate scaling for the Reynolds stresses. It is shown that this model is not only physically plausible but is also consistent with the available data.