  We determine an approximate expression for the O(alpha_s^3) contribution
chi_2 to the kernel of the BFKL equation, which includes all collinear and
anticollinear singular contributions. This is derived using recent results on
the relation between the GLAP and BFKL kernels (including running-coupling
effects to all orders) and on small-x factorization schemes. We present the
result in various schemes, relevant both for applications to the BFKL equation
and to small-x evolution of parton distributions.
