A composition comprising: i) at least one anhydrous bleaching composition in paste form comprising at least one peroxygenated salt, at least one alkaline agent, and from 15% to 35% by weight of at least one inert organic liquid, and ii) at least one oxidizing composition in the form of a hydrogen peroxide oil-in-water emulsion comprising at least one surfactant chosen from nonionic and anionic surfactants and at least one copolymer comprising at least one hydrophobic unit and at least one unit derived from at least one ethylenically unsaturated monomer comprising at least one sulphonic group, in free or partially or totally neutralized form. The composition may, for example, be used for bleaching human keratin fibers, such as hair. Also disclosed is a process for preparing a ready-to-use composition, comprising mixing, before use, the bleaching composition with the oxidizing composition.