A positive-working lithographic printing plate precursor for infrared laser is provided that includes, layered sequentially above a support, a lower layer and an upper layer, the lower layer and/or the upper layer including an infrared absorbing agent, either the lower layer comprising an alkali-soluble group-containing graft copolymer or the upper layer comprising a sulfonamide group-, active imide group-, and/or amide group-containing graft copolymer, and the graft copolymer being a polyurethane resin having as a graft chain an ethylenically unsaturated monomer-derived constitutional unit. There is also provided a process for making a lithographic printing plate, the process including in sequence an exposure step of imagewise exposing by means of an infrared laser the positive-working lithographic printing plate precursor for infrared laser and a development step of developing using an aqueous alkali solution with a pH of 8.5 to 10.8.