The physical nature of psi phenomena such as telepathy is an important problem in present science of consciousness. Scientists have basically confirmed the existence of telepathy phenomena through many strict experiments. Then can modern science (e.g. quantum theory) provide a scientific explanation for telepathy phenomena? In this paper, we will seek the possible quantum nature of telepathy from both theoretical and experimental aspects, and will present a primary quantum model of telepathy process. It is well known that even though present quantum theory permits the existence of quantum nonlocality, it does forbid the realization of nonlocal communication or quantum superluminal communication (QSC). However, the usual no-go theorems don’t consider the possible active role of consciousness during quantum measurement process. In a recent paper (see Found. Phys. Lett, 17(2), 167-182), it has been demonstrated that a proper combination of quantum process and conscious perception will permit the distinguishability of nonorthogonal quantum states, and further result in the realization of QSC or nonlocal communication. This is called the QSC principle. On the basis of the QSC principle, we propose a primary theoretical model of telepathy process. According to the model, the telepathy process mainly includes three phases. The first phase is to form the quantum entanglement state of brains, the second phase is to hold the entanglement state of brains, and the third phase is to collapse the entanglement state of brains. When the entanglement state of brains is collapsed by a certain measurement on one of the subjects, the brain states of both subjects turn to be definite states from entanglement state, and the other subject will perceive the change at a distance according to the QSC principle. When in the entanglement state or superposition state, no definite perception relating to the state exists, whereas when the superposition state collapses into a definite state, a definite perception relating to the collapse state appears. Then the telepathy between the subjects may appear. It should be stressed that, even though the above quantum model may in principle provide a primary scientific explanation of telepathy phenomena, there are still some left technical problems such as the expression of high-level telepathy information etc. Lastly, in order to test the QSC principle and the above quantum model of telepathy, some feasible schemes of quantum perception experiments and perception entanglement experiments are further proposed on the basis of present technology. We urge that such quantum perception experiments need to be conducted as soon as possible. If the experiment results are positive, they will have far-reaching influence on the present science of consciousness and psi research, and will help to bridge the gap between the parapsychology and present science.