The low-lying excited states in 132,134 Ba isotopes have been studied with high-resolution (p,t) reactions. The experiments were performed at the Munich Q3D spectrograph with a 25-MeV proton beam and the 1.5-m-long focal plane detector. The high-resolution triton spectra allowed the observation of levels up to ∼4 MeV. The experimental results revealed 75 excited states in 134 Ba and 79 in 132 Ba, many of them observed for the first time. The measured angular distributions compared with distorted-wave Born approximation calculations allowed spin assignments for these levels in most cases. The systematics of the monopole and quadrupole two-neutron transfer strengths is compared with the prediction of the interacting boson approximation model. The results indicate a transitional structure in 132 Ba and 134 Ba and contribute additional evidence in favor of a description between the U(5) and O(6) symmetries of the model.