We have investigated the nucleosynthesis of sodium (23Na) in stars of masses M = 5-19 M⊙ having solar-like initial chemical composition. The values obtained for the Na excess after the first dredge-up phase are in close agreement with recent observations suggesting a moderate Na excess in F-type supergiants. We also found a positive correlation between the overabundance factors [N/H] and [Na/H] which seems to indicate that Na enrichment originates from the Ne-Na cycle operating simultaneously with the CNO tri-cycle in these stars. We emphasize that our results were obtained on the basis of standard physical assumptions in the stellar model calculations, but with updated reaction rates for the reactions involved in the Ne-Na cycle which are presented in this work.