The peritectic reaction consists in the growth of the peritectic solid phase along the metastable interface between the primary solid phase and the liquid phase. We study theoretically the two-dimensional isothermal peritectic reaction in the limit of small undercoolings using the boundary-integral technique. First, we focus on the case where the liquid phase occupies a semi-infinite space and the peritectic phase presents a finger-like shape. Secondly, we investigate the case where the growth takes place in a channel of liquid phase and the peritectic phase fills the whole channel as a product of the transformation. It is found that for a critical channel width, the velocities of the channel filling and the finger-like solutions are equal. For smaller (larger) widths, the channel filling (finger-like) solution has a larger velocity and is likely to be predominant. This confirms some previously reported qualitative results of time-dependent phase-field calculations (Boussinot et al., 2010) [5]. We also discuss the relevance of our study to directional solidification experiments.