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We describe the B0 and Bs0 decays into J/ψ f0(500) and J/ψ f0(980) by taking into account the dominant process for the weak decay of B0 and Bs0 into J/ψ and a qq¯ component. After hadronization of this qq¯ component into pairs of pseudoscalar mesons we obtain certain weights for the meson–meson components and allow them to interact among themselves. The final state interaction of the meson–meson components, described in terms of chiral unitary theory, gives rise to the f0(980) and f0(500) resonances and we can obtain the π+π− invariant mass distributions after the decay of the resonances, which allows us to compare directly to the experiments. We obtain ratios of J/ψ f0(980) and J/ψ f0(500) for each of the B decays in quantitative agreement with experiment, with the f0(980) clearly dominant in the Bs0 decay and the f0(500) in the B0 decay.