- We investigate the effects of weakly and strongly stratified turbulent convection on the mean effective Lorentz force, and especially on the mean effective magnetic pressure. Earlier studies with isotropically forced non-stratified and stratified turbulence have shown that the contribution of the turbulence to the mean magnetic pressure is negative for mean horizontal magnetic fields that are smaller than the equipartition strength, so that the effective mean magnetic pressure that takes into account the turbulence effects can be negative. Compared with earlier cases of forced turbulence with an isothermal equation of state, we find that the turbulence effect is similar to or even stronger in the present case of turbulent convection. This is argued to be due to the anisotropy of turbulence in the vertical direction. Another important difference compared with earlier studies is the presence of an evolution equation for the specific entropy. Mean-field modelling with entropy evolution indicates that the negative effective magnetic pressure can still lead to a large-scale instability which forms local flux concentrations, even though the specific entropy evolution tends to have a stabilizing effect when applied to a stably stratified (e.g. isothermal) layer. It is argued that this large-scale instability could be important for the formation of solar large-scale magnetic structures such as active regions.