In this article, several fuzzy flexible distributions are used to model fuzzy income data, and then we estimate the upper and lower bounds of the Gini coefficient. Estimates are obtained using the maximum likelihood method. We have conducted a simulation study to evaluate the performance of the proposed bounds. The simulation results showed that the introduced bounds have performed well and include the Gini coefficient. The data used in this article is related to Iranian household income data in both urban and rural areas, which was taken from the website of the Iran Statistics Center, https://www.amar.org.ir, for the years 2011,2014,2017,2020 and 2021.