  In this paper we study the phenomenological implications of the one zero
textures of low energy neutrino mass matrices in presence of a sterile
neutrino. We consider the 3+1 scheme and use the results from global fit for
short baseline neutrino oscillation data which provides the bounds on the three
additional mixing angles. We find that the mass matrix elements $m_{\alpha
\beta}$ ($\alpha, \beta = e, \mu, \tau$) involving only the active states can
assume vanishing values in the allowed parameter space for all the mass
spectrum. Among the mass matrix elements connecting the active and sterile
states, $m_{es}$ and $m_{\mu s}$ can become small only for the quasi-degenerate
neutrinos. The element $m_{\tau s}$ on the other hand can vanish even for lower
values of masses since the 3-4 mixing angle only has an upper bound from
current data. The mass matrix element ($m_{ss}$) involving only the sterile
state stays $\sim$ $\mathcal{O}$(1) eV in the whole parameter region. We study
the possible correlations between the sterile mixing angles and the Majorana
phases to give a zero element in the mass matrix.
