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The present investigation shows quantitative results for the peritectic phase transformation of FeMn alloys utilizing phase-field simulations in 1-D and 2-D. The phase-field method used was based on an adaptation of the proposal of Folch and Plapp [Phys. Rev. E, 2005, 72, 011602] for the eutectic reaction. The two stages of peritectic phase transformation, the peritectic reaction and the peritectic transformation, were investigated numerically utilizing this phase-field approach. The evolution of the phases was quantitatively analyzed during the peritectic transformation and the fractions of the phases at the end of the solidification were compared with the thermodynamic equilibrium, defined by the phase diagram, for the case of 1-D simulation with peritectic concentration. An assessment of the behavior of the concentration gradient in the γ-phase (the peritectic phase) through time was also carried out and a mathematical function which describes the γ-phase thickness evolution was defined. Finally, 2-D simulations were performed to clearly identify the two stages of the peritectic phase transformation. The obtained results show two main facts: (1) the numerical model is able to simulate quantitatively this phase transformation; and, (2) this numerical tool can be utilized for investigating quantitatively some aspects (normally determined indirectly) that are difficult to be determined by direct measurements in experimental works.