Classical theories of liquid-crystalline materials are used to develop a new model that describes the thermodynamic stability of nematic liquid-crystal cylindrical fibers that arise in the fabrication of in-situ liquid crystal polymer (LCP) composites. The thermodynamic model identifies the new contributions to elastic storage due to the nematic orientational order present in LCP fibers. It is shown that the additional nematic surface and bulk elastic storage mechanisms tend to promote fiber stability when compared with isotropic fibers. The theory predicts that elastic storage due to orientational deformations within the fiber may be able to overcome the classical capillary (Rayleigh) instability present in isotropic fibers. The parametric conditions that lead to fiber stability are smaller fibers, low interfacial tensions, and large nematic elastic constants. Nevertheless, using estimates typical of the actual in situ LCP-polymer composites, it is found that LCP fibers are unstable and will break up, in agreement with existing experimental studies on the stability of liquid-crystalline polymer fibers in a thermoplastic elastomeric matrix when subjected to annealing at high temperatures.