We demonstrate the possible existence of a new class of superconducting Josephson junctions (JJs) named field-induced Josephson junctions (FIJJs). One representative is a junction made by placing a ferromagnetic strip on the top of a superconducting strip, which we study in this work. We obtain a possible transition between one regime of an FIJJ, which is the quasi-tunneling weak link Josephson junction (similar in certain aspects to an S-F-S Josephson junction, but with different boundary conditions), and the second regime of an FIJJ, which is the weak link constriction (two superconductors connected by a narrow superconducting link) as a function of the thickness of the superconducting strip, magnitude of magnetization, and temperature. We use the generalized Ginzburg-Landau (GL) equations derived from the extended Hubbard model in order to determine some properties of the new class of Josephson junction. In this paper, we perform the computation for a first type of superconductor, although both types of superconductor can be used to create an FIJJ. Further directions of studies of the indicated class of Josephson junction are also described.