The effect of 10 at % Mn introduced into the classic FINEMET (Fe73.5Cu1Nb3Si13.5B9) at the expense of Fe on the magnetic properties, magnetic anisotropy, and structure, which are formed after nanocrystallizing annealing at 520°C for 10 min to 4 h in the presence of tensile stresses σ = 200 MPa and in the absence of them, is studied. It is shown that, for the Fe63.5Mn10Cu1Nb3Si13.5B9 alloy, like for the classic FINEMET, the annealing in the presence of tensile stresses results in inducing the transverse magnetic anisotropy; however, the constant of induced magnetic anisotropy decreases by 4 times. The coercive force of the Mn-containing alloy increases as the time of annealing both in the presence of tensile stresses and without them increases, whereas the coercive force of FINEMET is almost unchanged. It is shown that the Mn-containing alloy subjected to 10-min treatments already is characterized by the presence of the borides along with the α-Fe(Si, Mn) solid solution and Fe3Si phase. This leads to changing the ratio of volume fractions of structure constituents with the negative and positive magnetostriction, which is likely to cause the decrease in the induced magnetic anisotropy constant of the Mn-containing alloy. The average grain size in the Mn-containing alloy is shown to increase as the time of treatments increases; this fact, along with the formation of borides, leads to the increase in the coercive force.