Nuclear reactions present an interesting case for studies of the time evolution of entanglement between complex quantum systems. In this work, the time-dependent nuclear density-functional theory is employed to explore entanglement in multinucleon transfer reactions. As an illustrative example, for the reaction Ca40+Pb208 at Elab=249MeV, in the interval of impact parameters 4.65-7.40fm, and the relativistic density-functional PC-PK1, we compute the von Neumann entropies, entanglement between fragments, nucleon-number fluctuations, and Shannon entropy for the nucleon-number observable. A simple linear correlation is established between the entanglement and nucleon-number fluctuation of the final fragments. The entanglement between the fragments can be related to the corresponding excitation energies and angular momenta. The relationship between the von Neumann entropy and the Shannon entropy for the nucleon-number observable is analyzed, as well as the time evolution of the entanglement (nucleon-number fluctuation). The entanglement is also calculated for a range of incident energies and it is shown how, depending on the impact parameter, the entanglement increases with the collision energy.