The ν-zeros of the Bessel functions of purely imaginary order are examined for fixed argument x > 0. In the case of the modified Bessel function of the second kind Kiν (x), it is known that it possesses a countably infinite sequence of real ν-zeros described by νn ∼ πn/ log n as n → ∞. Here we apply a unified approach to determine asymptotic estimates of the ν-zeros of the modified Bessel functions Liν (x) ≡ Iiν(x) + I−iν(x) and Kiν(x) and the ordinary Bessel functions Jiν (x)±J−iν(x). Numerical results are presented to illustrate the accuracy of the expansions so obtained.