For efficient analysis of non-stationary signals, such as radar, sonar, speech, music, or general audio, timefrequency (TF) representations are required. They allow performing non-stationary filtering, ie extracting nonstationary "components" in signals. However, ad-hoc procedures are frequently used, whose accuracy is difficult to assess and control. In this contribution, we are interested in such time-frequency filtering, which we define as follows: the attenuation of a given, small region Ω in the time-frequency plane. This problem, addressed in a recent publication, is referred to as time-frequency fading (TFF). We address a local version of TFF, called LTFF, that only exploits a neighborhood Ω of the time-frequency region of interest Ω, which is supposed to be much smaller than the full time-frequency domain. This problem will be referred to as local time-frequency fading. We propose a method for determining such a neighborhood Ω. The local TFF problem is then solved using a variational approach, that minimizes a suitable quadratic objective function. Thus, there is a closed-form solution involving Gabor multipliers.