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Accurate and controlled movement of small, untethered objects within soft tissues has many potential applications in medical robotics. While medium reaction forces due to slow movement of solid objects in viscoelastic fluids are well-known, such forces have received much less attention in soft media and tissues where the movement is accompanied by highly non-linear and history dependent phenomena. This paper develops a model of such forces for spherical solids. The reaction forces are investigated experimentally in the limit when the spherical solid moves at only a small fraction of its diameter per second. A mathematical model consistent with observations is proposed. The key element of the model is the history-dependent nature of the medium reaction force. A method of the model parameter identification is described, and its experimental implementation is demonstrated in gels that simulate soft tissues. In the experiments, known magnetic forces are employed as the external forces to drag a permanent magnet sphere inside Agarose gel phantom, and video tracking assisted by template matching calculations is used to accurately track the sphere translation. Numerical simulations of the model illustrate results that are consistent with observations.