These systems can be considered as a model for the spread of the epidemic in the population. In addition, systems with linearly increasing delay describe the dynamics of the information server, mixing tank, the process of formation of traffic jams on the ring road, etc. For the study, the concept of an average system is introduced. This approach allows us to reduce the analysis of the Lyapunov stability problem of the zero solution of the original system to the investigation of the zero solution of the averaged system. Sufficient conditions for stationary system stability are formulated. Then the application of Razumihin's approach to the study of stability original system is used. The Lyapunov function is constructed. As a result, new sufficient conditions for the asymptotic stability of the zero solution of nonstationary homogeneous systems with a linearly increasing time delay are obtained. These conditions are the generalization of well-known results for the linear systems with a linearly increasing time delay.