The classical Lur'e problem consists of finding conditions for absolute stability of a linear system with a nonlinear feedback contained within a prescribed sector. Most of the results obtained on this problem are based on the frequency domain or Lyapunov functions methods which are applied to systems with a time-invariant or periodic linear block. This paper develops a new approach providing a sufficient stability criterion for systems with time-variable coefficients, which is expressed in the transfer function of the linear block and the sector margins of the nonlinear block. The systems for which this criterion is precise are found. It is shown that stability of a system with a sign-constant transfer function is guaranteed by stability of the system with a limit linear feedback (so that, for such systems, the famous Aizerman conjecture is true). This, in particular, is the case for systems with a linear block consisting of an arbitrary number of first order time-dependent links. As an example, the stability criterion is applied to a second order system for which the obtained results are contrasted with ones delivered by the Popov criterion.