Ore extensions provide a way of constructing new algebras from preexisting ones, by adding a new indeterminate subject to commutation relations. A recent generalization of this concept is that of double Ore extensions. On the other hand, Hopf algebras are algebras which possess a certain additional dual structure. The problem of extending a Hopf algebra structure through an Ore extension has been discussed in a recent paper by Brown, O'Hagan, Zhang and Zhuang, of which we present the main result. We then address the same problem but concerning double Ore extensions over a field and how it ties to the general case. We split the cases which are possible or not with respect to the data that determines the double Ore extension.