Nowadays, there is a need for new sources of noble metals due to their dwindling natural resources. This paper presents studies on the sorption of noble metals such as Au(III), Pt(IV), Pd(II) and Rh(III) from model chloride solutions on a newly prepared Amberlite XAD-16–Aliquat 336 sorbent. A “warm impregnation” method without the use of toxic organic solvents was applied to impregnate the polymer matrix. The influence of such factors as hydrochloric acid concentration, sorbent mass and phase contact time was investigated. Kinetic as well as adsorption isotherm studies were carried out. The sorption capacity of the synthesized sorbent was Au(III)—94.34 mg/g, Pt(IV)—45.35 mg/g and Pd(II)—46.03 mg/g. Based on thermodynamic considerations, their sorption proved to be endothermic, as the values of ΔH° > 0. Sorption was spontaneous and favourable (ΔG° < 0). After leaching the RAM module, there was obtained a real solution, in which the metal contents were determined: 38.10 mg/g of gold and 1.76 mg/g of palladium. Totals of 99.9% of gold and 45.4% of palladium were removed from the real leaching solution, with other elements in the solution.