Data integration is now common practice in official statistics and involves an increasing number of sources. When using multiple sources, an objective is to assess the unknown size of the population. To this aim, capture-recapture methods are applied. Standard capture-recapture methods are based on a number of strong assumptions, including the absence of errors in the integration procedures. However, in particular when the integrated sources were not originally collected for statistical purposes, this assumption is unlikely and linkage errors (false links and missing links) may occur. In this article, the problem of adjusting population estimates in the presence of linkage errors in multiple lists is tackled; under homogeneous linkage error probabilities assumption, a solution is proposed in a realistic and practical scenario of multiple lists linkage procedure.