We demonstrate the presence of ferromagnetic (FM) fluctuations in the superconducting and nonsuperconducting heavily overdoped regimes of high-temperature superconducting copper oxides, using (Bi,Pb)_{2}Sr_{2}CuO_{6+δ} (Bi-2201) single crystals. Magnetization curves exhibit a tendency to be saturated in high magnetic fields at low temperatures in the heavily overdoped crystals, which is probably a precursor phenomenon of a FM transition at a lower temperature. Muon spin relaxation detects the enhancement of spin fluctuations at high temperatures below 200 K. Correspondingly, the ab-plane resistivity follows a 4/3 power law in a wide temperature range, which is characteristic of metals with two-dimensional FM fluctuations due to itinerant electrons. As the Wilson ratio evidences the enhancement of spin fluctuations with hole doping in the heavily overdoped regime, it is concluded that two-dimensional FM fluctuations reside in the heavily overdoped Bi-2201 cuprates, which is probably related to the decrease in the superconducting transition temperature in the heavily overdoped cuprates.