Accurately and efficiently addressing the multiple source localization (MSL) problem in urban environments, particularly designing a general method adaptable to an arbitrary number of sources, plays a crucial role in various fields such as cognitive radio (CR). Existing methods either fail to effectively utilize received signal strength (RSS) information without redundancy or lack generalizability to an arbitrary number of sources. In this work, we propose the Local Radio Map-Aided Multiple Source Localization Framework (LRM-MSL), which is a general method capable of handling an arbitrary number of sources. First, this framework constructs a local radio map that retains only the RSS information around the sources and binarizes it. Then, the connected component analysis tool is applied to the binarized map, which implements multi-source separation, transforming the MSL problem into a series of single-source localization (SSL) tasks. Finally, we design a numerical coordinate regression network to perform the SSL tasks. Since there is no publicly available RSS dataset for MSL, we construct the VaryTxLoc dataset to evaluate the performance of LRM-MSL. Experimental results demonstrate that LRM-MSL is an accurate and effective method, outperforming state-of-the-art approaches. Our code and dataset can be downloaded from https://github.com/hereis77/LRM-MSL.