A new lattice Bhatnagar-Gross-Krook (LBGK) model for a class of the generalized Burgers equations is proposed.It is a general LBGK model for nonlinear Burgers equations with source term in arbitrary dimensional space. The linear stability of the model is also studied. The model is numerically tested for three problems in different dimensional space, and the numerical results are compared with either analytic solutions or numerical results obtained by other methods.Satisfactory results are obtained by the numerical simulations.