In　this　paper,　we　introduce　the　notion　of　(*)-serial　coalgebras　which　is　a　generalization　of　serial　coalgebras.　We　investigate　the　properties　of　(*)-serial　coalgebras　and　their　comodules,　and　obtain　sufficient　and　necessary　conditions　for　a　basic　coalgebra　to　be　(*)-serial.