We present a general scheme for the calculation of the Rényi entropy of a subsystem in quantum many-body models that can be efficiently simulated via quantum Monte Carlo. When the simulation is performed at very low temperature, the above approach delivers the entanglement Rényi entropy of the subsystem, and it allows us to explore the crossover to the thermal Rényi entropy as the temperature is increased. We implement this scheme explicitly within the stochastic series expansion as well as within path-integral Monte Carlo, and apply it to quantum spin and quantum rotor models. In the case of quantum spins, we show that relevant models in two dimensions with reduced symmetry (XX model or hard-core bosons, transverse-field Ising model at the quantum critical point) exhibit an area law for the scaling of the entanglement entropy.