In this paper we are concerned with the global "nonnegative" approximate controllability property of a rather general semilinear heat equation with superlinear term, governed in a bounded domain Omega subset of R(n) by a multiplicative (bilinear) control in the reaction term like vu(x, t), where v is the control. We show that any nonnegative target state in L(2)(Omega) can approximately be reached from any nonnegative, nonzero initial state by applying at most three static bilinear L(infinity)(Omega)-controls subsequently in time. This result is further applied to discuss the controllability properties of the nonhomogeneous version of this problem with bilinear term like v(u(x, t) -theta(x)), where theta is given. Our approach is based on an asymptotic technique allowing us to distinguish and make use of the pure diffusion and/or pure reaction parts of the dynamics of the system at hand, while suppressing the effect of a (general) nonlinear term.