Soft grippers are gaining significant attention in the manipulation of elastic objects, where it is required to handle soft and unstructured objects, which are vulnerable to deformations. The crucial problem is to estimate the physical parameters of a squeezed object to adjust the manipulation procedure, which poses a significant challenge. The research on physical parameters estimation using deep learning algorithms on measurements from direct interaction with objects using robotic grippers is scarce. In our work, we proposed a trainable system which performs the regression of an object stiffness coefficient from the signals registered during the interaction of the gripper with the object. First, using the physics simulation environment, we performed extensive experiments to validate our approach. Afterwards, we prepared a system that works in a real-world scenario with real data. Our learned system can reliably estimate the stiffness of an object, using the Yale OpenHand soft gripper, based on readings from Inertial Measurement Units (IMUs) attached to the fingers of the gripper. Additionally, during the experiments, we prepared three datasets of IMU readings gathered while squeezing the objects—two created in the simulation environment and one composed of real data. The dataset is the contribution to the community providing the way for developing and validating new approaches in the growing field of soft manipulation.