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GW190521 is a merger of two black holes (BHs), wherein at least one BH lies within the pair-instability (PI) mass gap, and it is difficult to form because of the effects of PI supernovae (PISNe) and pulsational PI (PPI). In this study, we examined the formation of GW190521-like BH-BHs under Population (Pop) III environments by binary population synthesis calculations. We reveal that convective overshooting in stellar evolution strongly affects the formation of GW190521-like BH-BHs. A model with a small overshoot parameter (similar to GENEC) can form GW190521-like BH-BHs. The derived merger rate is $4 \times 10^{-2}$ ${\rm yr}^{-1}$ ${\rm Gpc}^{-3}$ at a redshift of $\sim 0.82$, which is comparable to the merger rate of GW190521-like BH-BHs inferred by gravitational wave (GW) observations. In this model, a $\sim 90~M_\odot$ star collapses to form a $\sim 90~M_\odot$ BH by avoiding PPI and PISN even if it is a member of a binary star. This is because it expands up to $10^2~R_\odot$, and lose only little mass through binary evolution. However, a model with a large overshoot parameter (similar to Stern) cannot form GW190521-like BH-BHs at all. Thus, we cannot conclude that a Pop~III binary system is the origin of GW190521 because determination of the overshoot parameter involves highly uncertain. If a Pop~III binary system is the origin of GW190521, the merger rate of BH-BHs including a $100-135~M_\odot$ BH is substantially smaller than that of GW190521-like BH-BHs. This will be assessed by GW observations in the near future.