We explore possible signatures for observing Majorana fermions in the tunneling spectroscopy of high-Tc superconductors. It is shown that due to the Rashba spin-orbit interaction (αR) generally introduced by contact electrode, in addition to the Heisenberg spin exchange interaction, the Dzyaloshinskii-Moriya and spin dipole-dipole interactions are induced. As a result, p-wave superconductivity is induced with the gap-function d vector being not aligned with the internal magnetic field of the spin-orbit interaction. For typical strength of the Rashba interaction, the induced p wave is weak. Hence, the resulting superconductor is still gapless and is not a topological superconductor. However, we find that the ground state undergoes a phase transition to a topological gapless phase with each nodal point originated from pure d wave being split into two stable nodal points characterized by the symmetry class DIII. Due to the splitting nodal structure, zero-energy Majorana modes always exist for any interfaces that are not exactly in (100) or (010) directions. Hence, for general interfaces, existence of Majorana modes is a robust feature. In addition, due to the nonaligned d vector, for (110) interfaces in which d wave is subdominant to p wave, there exist sizable dispersive Majorana edge states. Our results indicate that due to the presence of these Majorana modes, a small plateau in tunneling spectrum near zero-bias peak would be induced. Furthermore, zero-energy Majorana modes result in 4π periodicity in typical SIS′ junctions with difference in orientations of S and S′ being within 21'-39' for αR=0.05-0.3 eV. As a result, it is easy for a π ring in tricrystal experiments to hold Majorana fermions and exhibit periods of two flux quanta in external magnetic field. These phenomena may have been already observed in experiments and their connections to experimental results are discussed.