  We present a Bethe approximation to study lattice models of linear polymers.
The approach is variational in nature and based on the cluster variation method
(CVM). We focus on a model with $(i)$ a nearest neighbor attractive energy
$\epsilon_v$ between pair of non--bonded monomers, $(ii)$ a bending energy
$\epsilon_h$ for each pair of successive chain segments which are not
collinear. We determine the phase diagram of the system as a function of the
reduced temperature $t=\frac{T}{\epsilon_v}$ and of the parameter
$x=\frac{\epsilon_h}{\epsilon_v}$. We find two different qualitative behaviors,
on varying $t$. For small values of $x$ the system undergoes a $\theta$
collapse from an extended coil to a compact globule; subsequently, on
decreasing further $t$, there is a first order transition to an anisotropic
phase, characterized by global orientational order. For sufficiently large
values of $x$, instead, there is directly a first order transition from the
coil to the orientational ordered phase. Our results are in good agreement with
previous Monte Carlo simulations and contradict in some aspects mean--field
theory. In the limit of Hamiltonian walks, our approximation recovers results
of the Flory-Huggins theory for polymer melting.
