Recently,　Hattori–Lagha　established　the　global　existence　and　asymptotic　behavior　of　the　solutions　for　a　three-dimensional　compressible　chemotaxis　system　with　chemoattractant　and　repellent　(Hattori　and　Lagha　in　Discrete　Contin.　Dyn.　Syst.　41(11):5141–5164,　2021).　Motivated　by　Hattori–Lagha’s　work,　we　further　investigated　the　optimal　time-decay　rates　of　strong　solutions　with　small　perturbation　to　the　three-dimensional　Keller–Segel　system　coupled　to　the　compressible　Navier–Stokes　equations,　which　models　for　the　motion　of　swimming　bacteria　in　a　compressible　viscous　fluid.　First,　we　reformulate　the　system　into　a　perturbation　form.　Then　we　establish　a　prior　estimates　of　solutions　and　prove　the　existence　of　the　global-in-time　solutions　based　on　the　local　existence　of　unique　solutions.　Finally,　we　will　establish　the　optimal　time-decay　rates　of　the　nonhomogeneous　system　by　the　decomposition　technique　of　both　low　and　high　frequencies　of　solutions　as　in　(Wang　and　Wen　in　Sci.　China　Math.,　2020,　https://doi.org/10.1007/s11425-020-1779-7).　Moreover,　the　decay　rate　is　optimal　since　it　agrees　with　the　solutions　of　the　linearized　system.　©　2022,　The　Author(s).