  For each rational number $p/q\in (1/2,\sqrt 2/2)$ one can construct an
$\mathbb S^1$-equivariant minimal torus in $\mathbb S^3$ called Otsuki torus
and denoted by $O_{p/q}$. The Lawson's bipolar surface construction applied to
$O_{p/q}$ gives a minimal torus $\widetilde O_{p/q}$ in $\mathbb S^4$. In this
paper we give upper and lower bounds on the Morse index and the nullity of
these tori for $p/q$ close to $\sqrt 2/2$. We also state a numerically assisted
conjecture concerning the general case.
