One of the most important pieces of equipment in power systems is the power transformer. The main task of transformers is to change the voltage level in the system to deliver the generated energy to the final consumers. Power transformers are the main components of power systems because, without them, power transmission from the power plants to the consumers is not practically possible. Therefore, the importance of power transformers makes their protection a crucial issue. Differential relay is the relay most commonly used to protect power transformers throughout the world. Despite their many advantages, these types of relays may operate incorrectly during the magnetizing inrush current of the transformer and cause the healthy transformer to be separated from the power network. Thus, using some appropriate techniques to cope with this problem is necessary for power transformer protection. In a power network, the sampling rate can significantly impact the performance of the protection system. In this case, as the sampling rate increases, the computational complexity also rises. On the other hand, if the sampling rate decreases, it can reduce the accuracy of the protection system. Therefore, this paper presents a new method based on adaptive sampling and Hilbert transform to discriminate between inrush current and internal faults in a power transformer. The proposed method is also capable of detecting sympathetic inrush current. Besides, this method precisely detects various fault types and has an accurate performance when the current transformer is saturated, or the signal is noisy. The proposed method has been tested on a 230/63 kV transformer. The simulation results demonstrate that the proposed method has an appropriate performance during normal and sympathetic inrush currents. In addition, the proposed technique can distinguish between all fault types and inrush current in power transformers.