In this article the strong observability conditions, necessary and sufficient conditions originally stated for the state reconstruction of linear systems with unknown inputs, are extended to the class of linear time varying singular systems. This novel local (in a time interval of interest) conditions are given under a constant rank assumption. A method to reconstruct the state in finite time in linear time-varying singular systems despite the presence of unknown inputs is presented. The proposed observer consists of two parts, one part that ensures the asymptotic convergence of the estimation errors to a bounded region around the origin and another one that ensures the finite-time convergence of the state estimates to their actual values. (C) 2018 Elsevier Ltd. All rights reserved.