It is well known that the presence of continuous fibres in SiC/Ti composites leads to a high mechanical anisotropy of the composite between the longitudinal and the two transverse directions. But it is also possible that the crystallographic texture of the matrix may lead to a non-negligible anisotropy of the mechanical properties of the composite. The crystallographic orientation of the matrix grains was determined using the Electron BackScattering Diffraction technique. A local texture of the matrix of the composite is thus evidenced. Finite Element calculations are used to determine the stress field in the matrix resulting from an applied transverse loading. The representative mechanical quantities thus determined are discussed in relation with the fundamental mechanisms of plastic deformation of the matrix. Finally, the crystallographic texture of the matrix of the composite is taken into account in the numerical modellings using a three-scale model that combines crystal plasticity, a polycrystalline aggregate model and a periodic homogenization through a Finite Element unit cell for the composite analysis.