A multidimensional cosmological-type model with n Einstein factor spaces in the theory with l scalar fields and multiple exponential potentials is considered. The dynamics of the model near the singularity is reduced to a billiard on the (N−1)-dimensional Lobachevsky space HN−1, N = ls; n+l. It is shown that for n > 1 the oscillating behaviour near the singularity is absent and solutions have an asymptotic Kasner-like behavior. For the case of one scale factor (n = 1) billiards with finite volumes (e.g. coinciding with that of the Bianchi-IX model) are described and oscillating behaviour of scalar fields near the singularity is obtained.