  We calculate the magnetic susceptibility in the pseudogap phase of cuprates
using the two-dimensional $t-J$ model and the Mori projection operator
technique. In this phase, the Fermi arcs lead to a quasi-elastic incommensurate
magnetic response for low temperatures. The theory accounts for the
reorientation of the susceptibility maxima from the axial to the diagonal
direction occurring at small hole concentrations in La$_{2-x}$Sr$_x$CuO$_4$. A
small cusp of the hole dispersion near the Fermi level, which is connected with
the spin-polaron band, affords the growth of the maxima with decreasing
frequency. As in the superconducting phase, the susceptibility maxima have an
hourglass dispersion. The assumption of the phase separation into regions of
superconducting and pseudogap phases in underdoped cuprates explains the lack
of the superconducting gap in their susceptibility. The observed rapid change
of widths of the susceptibility maxima with frequency is related to the
contribution of the pseudogap phase. This contribution may also explain the
strengthening of the quasi-elastic response by modest magnetic fields and by
small impurity concentrations.
