Let X C P-d be an m-dimensional variety in d-dimensional complex projective space. Let k be a positive integer such that the combinatorial number ( m + k k ) is smaller than or equal to d . Consider the following interpolak tion problem: does there exist a variety Y C P-d of dimension strictly smaller than ( m + k k) , with X C Y , such that the tangent space to Y at a point p is an element of X is k equal to the k th osculating space to X at p , for almost all points p is an element of X ? In this paper we consider this question in the toric setting. We prove that if X is toric, then there is a unique toric variety Y solving the above interpolation problem. We identify Y in the general case and we explicitly compute some of its invariants when X is a toric curve.