We consider the following type of question: Given a partial proper d-edge coloring of the d-dimensional hypercube Qd, and lists of allowed colors for the non-colored edges of Qd, can we extend the partial coloring to a proper d-edge coloring using only colors from the lists? We prove that this question has a positive answer in the case when both the partial coloring and the color lists satisfy certain sparsity conditions. (C) 2020 Elsevier B.V. All rights reserved.