The invention provides a process for preparing dyed, polypropylene-based fibers. Polypropylene having a Melt Flow Index of 45 or less is combined with an additive blend to provide the polypropylene-based polymer composition for forming into fibers. The additive blend is a terpolymer of ethylene, acrylic or methacrylic ester and maleic anhydride or glycidyl methacrylate combined with an amine-terminated polyamide, in weight ratio 3:1 to 1:1. Fatty acid monoglyceride may also be used in the polymer composition to further improve dyeability. The polypropylene-based fibers are readily dyeable by contacting the fibers with an aqueous dispersion of a disperse dye, to provide fibers with high color intensity (deep hues) and high dye substantivity and color fastness, without the need to employ temperatures in excess of the atmospheric pressure boiling point of the dye dispersion.