  We consider a nonlocal version of the Nambu and Jona-Lasinio model. The
nonlocality is contained in the quark-antiquark bilinears of the four-quark
vertices as a form factor of the Gaussian type. The model has three parameters
which can be fixed in favour of the values of the pion mass, the pion decay
constant f_pi, and the current quark mass. The pi pi scattering amplitude is
obtained by calculating the quark box and the sigma-pole diagrams, where sigma
is the scalar isoscalar meson. It is shown that this amplitude satisfies the
well-known Weinberg relation. We obtain the s, p, d wave scattering lengths in
all isotopic channels and the s wave slope parameters. The results are in
satisfactory agreement with both phenomenological data and the basic
requirements of low-energy theorems, thus supporting to the form factor used.
