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The indentation of a rigid cylinder with a flat base into a viscoelastic layer situated without friction on a rigid base is considered. The interaction process consists of two stages: indentation of a cylinder at a constant speed to a certain depth and its further holding in this depth. The mechanical behavior of a viscoelastic layer is described by equations of linear viscoelasticity with relaxation functions of two types: exponential and power. The solution of the problem is based on the application of the Volterra principle and the use of an elastic solution constructed by the asymptotic “methods of large and small λ” that were developed by I. I. Vorovich and his students. On the basis of the solution obtained, the nature of the time variation in contact stresses and loads during the indentation of the cylinder has been studied for both relatively small and relatively large layer thicknesses. A comparison of the time variation of the studied quantities for two different relaxation functions and with the results of calculations based on a simplified one-dimensional model used to describe the behavior of a viscoelastic layer during indentation is performed.