Recently, we have developed a multiscale soft matter model for stem cells or primitive cells in general, aiming at improving the understanding of mechanotransduction mechanism of cells that is responsible for information exchange between cells and their extracellular environment. In this paper, we report the preliminary results of our research on multiscale modeling and simulation of soft contact and adhesion of cells. The proposed multiscale soft matter cell model may be used to model soft contact and adhesion between cells and their extracellular substrates. This model is a generalization of the Fluid Mosaic Model (Singer and Nicolson, 1972), or an extension of Helfrich’s liquid crystal membrane model (Helfrich, 1973). To the best of the authors’ knowledge, this may be the first time that a soft matter model is developed for cell contact and adhesion. Moreover we have developed and implemented a Lagrange type meshfree Galerkin formulation and the computational algorithm for the proposed cell model. Comparison study with experimental data has been conducted to validate the parameters of the model. By using the soft matter cell model, we have simulated the soft adhesive contact process between cells and their extracellular substrates. The simulation shows that the cell can sense substrate elasticity by responding in different manners from cell spreading motion to cell contact configurations.