  We consider the dependence of the MgB_2 superconducting critical temperature
on the pressure. Our model exploits the influence of the large polarons on the
band structure of the layered MgB_2 superconductor. Namely, the hole
Pekar-Froehlich polarons form quasi two-dimensional potential wells in the
boron plane which shift the positions of the sigma- and pi-bands. This energy
shift depends on the pressure and the Cooper pairing of the correlated
sigma-electrons happens inside polaron wells. The results obtained are as
follows: dT_c/dp = -\alpha (5.2 \pm 0.9) K/GPa or dT_c/dp = -\alpha (6.9\pm
1.1) K/GPa for a different choice of the Grueneisen parameter. Being compared
with known experimental data they give us a resonable interval for the value of
the Froehlich electron-phonon coupling constant: \alpha = 0.15 - 0.45.
