The main methods for producing genetically engineered cells use viral vectors for which safety issues and manufacturing costs remain a concern. In addition, selection of desired cells typically relies on the use of cytotoxic drugs with long culture times. Here, we introduce an efficient non-viral approach combining the Sleeping Beauty (SB) Transposon System with selective proliferation of engineered cells by chemically induced dimerization (CID) of growth factor receptors. Minicircles carrying a SB transposon cassette containing a reporter transgene and a gene for the F36VFGFR1 fusion protein were delivered to the hematopoietic cell line Ba/F3. Stably-transduced Ba/F3 cell populations with >98% purity were obtained within 1 week using this positive selection strategy. Copy number analysis by quantitative PCR (qPCR) revealed that CID-selected cells contain on average higher copy numbers of transgenes than flow cytometry-selected cells, demonstrating selective advantage for cells with multiple transposon insertions. A diverse population of cells is present both before and after culture in CID media, although site-specific qPCR of transposon junctions show that population diversity is significantly reduced after selection due to preferential expansion of clones with multiple integration events. This non-viral, positive selection approach is an attractive alternative for producing engineered cells.