In this paper, we introduce the bulk viscosity in the formalism of modified gravity theory in which the gravitational action contains a general function f(R,T) , where R and T denote the curvature scalar and the trace of the energy–momentum tensor, respectively, within the framework of a flat Friedmann–Robertson–Walker model. As an equation of state for a prefect fluid, we take p=(γ-1)ρ , where 0≤γ≤2 and a viscous term as a bulk viscosity due to the isotropic model, of the form ζ=ζ0+ζ1H , where ζ0 and ζ1 are constants, and H is the Hubble parameter. The exact non-singular solutions to the corresponding field equations are obtained with non-viscous and viscous fluids, respectively, by assuming a simplest particular model of the form of f(R,T)=R+2f(T) , where f(T)=αT ( α is a constant). A big-rip singularity is also observed for γ<0 at a finite value of cosmic time under certain constraints. We study all possible scenarios with the possible positive and negative ranges of α to analyze the expansion history of the universe. It is observed that the universe accelerates or exhibits a transition from a decelerated phase to an accelerated phase under certain constraints of ζ0 and ζ1 . We compare the viscous models with the non-viscous one through the graph plotted between the scale factor and cosmic time and find that the bulk viscosity plays a major role in the expansion of the universe. A similar graph is plotted for the deceleration parameter with non-viscous and viscous fluids and we find a transition from decelerated to accelerated phase with some form of bulk viscosity.