Motivated by the intrinsic non-Fermi-liquid behavior observed in the heavy-fermion quasicrystal Au51Al34Yb15, we study the low-temperature behavior of dilute magnetic impurities placed in metallic quasicrystals. We find that a large fraction of the magnetic moments are not quenched down to very low temperatures T, leading to a power-law distribution of Kondo temperatures P(TK)∼TKα-1, with a nonuniversal exponent α, in a remarkable similarity to the Kondo-disorder scenario found in disordered heavy-fermion metals. For α<1, the resulting singular P(TK) induces non-Fermi-liquid behavior with diverging thermodynamic responses as T→0.