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 = ζ 0 + ζ 1 H , where ζ 0 and ζ 1 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 + 2 f (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 ζ 0 and ζ 1 . 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.