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 6P_3/2 of neutral Tl. Based on the obtained theoretical and available experimental data, the nuclear magnetic moments of short-lived ¹⁹¹Tl^m and ¹⁹³Tl^m isotopes are predicted: μ₁₉₁ = 3.79(2) μ_N and μ₁₉₃ = 3.84(3) μ_N, respectively. Using theoretical and experimental data for the neutral Tl, the magnetic anomalies ²⁰⁵Δ²⁰³ for the 7S_1/2 state of the neutral Tl atom and the 1S_1/2 state of the hydrogen-like ion are also predicted.