  The gravitational wave measurements of spin-induced multipole moment
coefficients of a binary black hole system can be used to distinguish black
holes from other compact objects [N. V. Krishnendu et al., PRL 119, 091101
(2017)]. Here, we apply the idea proposed in [N. V. Krishnendu et al., PRL 119,
091101 (2017)] to binary systems composed of intermediate-mass and supermassive
black holes and derive the expected bounds on their Kerr nature using future
space-based gravitational wave detectors. Using astrophysical models of binary
black hole population, we study the measurability of the spin-induced
quadrupole and octupole moment coefficients using LISA and DECIGO. The errors
on spin-induced quadrupole moment parameter of the binary system are found to
be { $\leq 0.1$ for almost $3\%$ of the total supermassive binary black hole
population which is detectable by LISA whereas it is $\sim 46\%$ for the
intermediate-mass black hole binaries observable by DECIGO at its design
sensitivity.} We find that { errors on} {\it both} the quadrupole and octupole
moment parameters can be estimated to { be} $\leq 1$ for $\sim 2\%$ and $\sim
50\%$ {of the population} respectively for LISA and DECIGO detectors. { Our
findings suggest that a subpopulation of binary black hole events, with the
signal to noise ratio thresholds greater than 200 and 100 respectively for LISA
and DECIGO detectors, would permit tests of black hole nature to 10\%
precision.}
