The influence of the nuclear magnetization distribution effects on the hyperfine structure of electronic states of thallium atom is studied within the relativistic coupled cluster theory. Relative significance of these effects is demonstrated for the first excited electronic state 6P3/2 of neutral Tl. Based on the obtained theoretical and available experimental data, the nuclear magnetic moments of short-lived 191Tlm and 193Tlm isotopes are predicted: μ191 = 3.79(2) μN and μ193 = 3.84(3) μN, respectively. Using theoretical and experimental data for the neutral Tl, the magnetic anomalies 205Δ203 for the 7S1/2 state of the neutral Tl atom and the 1S1/2 state of the hydrogen-like ion are also predicted.