  We show that it is possible to construct a wide class of Lorentzian wormholes
in Eddington-inspired Born-Infeld gravity with a stress energy which does not
violate the weak or null energy condition. The wormholes exist in a certain
region of the parameter space. In fact, it is shown that there is a critical
value of a parameter defined in our work, below which we have wormholes. Above
the critical value, we have a regular black hole spacetime. We put a
restriction on the equation of state parameter $\alpha$ ($p_{\theta}=\alpha
\rho$) to have wormholes. We also put a lower limit on both the theory
parameter $|\kappa|$ and the throat radius, to restrict the tidal acceleration
(at the throat) below one Earth gravity. As a special case of our general
solution, we retrieve the wormhole supported by an electric field for a
charge-to-mass ratio greater than the critical value
$\left(\frac{Q}{M}\right)_c\approx 1.144$.
