Recent data from the PIONS@MAX-lab Collaboration, measuring the total cross section of the incoherent pion photoproduction reaction, γ d → π - p p , near threshold, have been used to extract the E 0 + multipole and total cross section of the reaction γ n → π - p , also near threshold. These are the first measurements of the reaction γ d → π - p p in the threshold region. The value of E 0 + is extracted through a fit to the deuteron data in a photoproduction model accounting for final-state interactions. The model takes an S -wave approximation for the elementary reaction γ n → π - p with E 0 + = const in the threshold region. The fit over all the 6 deuteron data points gives the value E 0 + = - 31.86 ± 0.8 (in units 10 - 3 / m π ). We explore the dependence of our results on the choice of data subsets included in the fit. The obtained values of E 0 + , for different subsets, have overlapping errors and agree with previous determinations.