  We calculate the bipartite von Neumann and second R\'enyi entanglement
entropies of the ground states of spin-1/2 dimerized Heisenberg
antiferromagnets on a square lattice. Two distinct dimerization patterns are
considered: columnar and staggered. In both cases, we concentrate on the
valence bond solid (VBS) phase and describe such a phase with the bond-operator
representation. Within this formalism, the original spin Hamiltonian is mapped
into an effective interacting boson model for the triplet excitations. We study
the effective Hamiltonian at the harmonic approximation and determine the
spectrum of the elementary triplet excitations. We then follow an analytical
procedure, which is based on a modified spin-wave theory for finite systems and
was originally employed to calculate the entanglement entropies of magnetic
ordered phases, and calculate the entanglement entropies of the VBS ground
states. In particular, we consider one-dimensional (line) subsystems within the
square lattice, a choice that allows us to consider line subsystems with sizes
up to $L' = 1000$. We combine such a procedure with the results of the
bond-operator formalism at the harmonic level and show that, for both dimerized
Heisenberg models, the entanglement entropies of the corresponding VBS ground
states obey an area law as expected for gapped phases. For both columnar-dimer
and staggered-dimer models, we also show that the entanglement entropies
increase but do not diverge as the dimerization decreases and the system
approaches the N\'eel--VBS quantum phase transition. Finally, the entanglement
spectra associated with the VBS ground states are presented.
