Many works within robust combinatorial optimisation consider interval-valued costs or constraints. While most of these works focus on finding a unique solution following a robust criteria such as minimax, a few consider the problem of characterising a set of possibly optimal solutions. This paper is situated within this line of work, and considers the problem of exactly enumerating the set of possibly optimal matroids under interval-valued costs. We show in particular that each solution in this set can be obtained through a polynomial procedure, and provide an efficient algorithm to achieve the enumeration.