The local stress-strain state in the vicinity of the crack tip in a composite is studied, taking into account the mechanical and geometric features of the nearest interface. The modeling of Mode I and II problems for a semi-infinite crack terminating normally at a nonideal interface in the bimaterial plane is considered. The constituents of the composite are assumed to be elastic, homogeneous, and isotropic. The intermediate zone between the constituents is modeled by interfacial conditions in the form: [σn] = 0, [u] = τrασn, where [u] and [σn] are jumps of the vectors of displacements and tractions along the interface. The diagonal matrix τ with nonnegative components and the parameter α ≥ 0 are defined by the mechanical and geometric characteristics of the intermediate zone, respectively. Thus, the case τ = 0 corresponds to the usual "ideal" contact conditions along the interface. Using the method of integral transformations, the corresponding problems are reduced to systems of functional equations, and later to systems of integral equations with fixed point singularities. The solvability of the systems of integral equations is proved and the asymptotics of their solutions is found. Based on these results, the local distributions of the displacements and stresses near the crack tip are obtained. It is shown that the interfacial parameters a and τ greatly influence the stress not only qualitatively (the character of the stress singularity near the crack tip changes), but also quantitatively (number of singular terms in the asymptotics increases). The graphs illustrating these results are presented as the values of the interfacial parameters a and τ, as well as the ratio of the shear moduli μ0/μ1 of the constituents.