A fully Eulerian approach to predict fluids-membrane behaviours is presented in this paper. Based on the numerical model proposed by Ii et al. (2012), we present a sharp methodology to account for the jump conditions due to hyperelastic membranes. The membrane is considered infinitely thin and is represented by the level set method. Its deformations are obtained from the transport of the components of the left Cauchy-Green tensor throughout time. Considering the linear or a hyperelastic material law, the surface stress tensor is computed and gives the force exerted by the membrane on the surrounding fluids. The membrane force is taken into account in the Navier-Stokes equations as jump conditions on the pressure and on the velocity derivatives by imposing suitable singular source terms in cells crossed by the interface. To prevent stability issues, an extension algorithm has been developed to remove the normal derivatives of the scalar fields specific to the membrane. In particular, a subcell resolution at the interface of the extrapolated variable is proposed for increasing the accuracy of the extension algorithm. These improvements are validated by comparing our numerical results with benchmarks from the literature. Moreover, a new benchmark is proposed for fluids with both different viscosities and different densities to target applications where a gas and a liquid phase are separated by a membrane.