  In this work, we intend to explore wormhole geometries in the framework of
$f(R,L_m)$ gravity. We derive the field equations for the generic $f(R,L_m)$
function by assuming the static and spherically symmetric Morris-Thorne
wormhole metric. Then we consider two non-linear $f(R,L_m)$ model,
specifically, $f(R,L_m)=\frac{R}{2}+L_m^\alpha$ and
$f(R,L_m)=\frac{R}{2}+(1+\lambda R)L_m$, where $\alpha$ and $\lambda$ are free
model parameters. We obtain the wormhole solutions by assuming three cases,
namely, a linear barotropic EoS, anisotropic EoS, and isotropic EoS
corresponding to model I. We observe that for both barotropic and anisotropic
cases, the corresponding wormhole solutions obey the flaring-out condition
under asymptotic background, while for the isotropic case, the shape function
does not follow the flatness condition. Also, we find that the null energy
condition exhibits negative behavior in the vicinity of the throat. Further, we
consider two different shape functions to investigate the behavior of model II.
We find some constraints on the model parameter for which the violation of the
null energy condition exhibits. Finally, we employ the volume integral
quantifier to calculate the amount of exotic matter required near the wormhole
throat for both models. We conclude that the modification of standard GR can
efficiently minimize the use of exotic matter and provide stable traversable
wormhole solutions.
