Summary: Let \(M\) be a subset of a (real) linear space that is closed with respect to the sum of vectors and the product by nonnegative scalars. An asymmetric seminorm on \(M\) is a nonnegative and subadditive positively homogeneous function \(q\) defined on \(M\). Moreover, \(q\) is an asymmetric norm if, in addition, for every nonzero element \(X\) such that \(-x\) belongs to \(M\), \(q(x)\) or \(q(-x)\) are different from zero. Consider the linear expansion \(X\) of \(M\). In this paper, we characterize when \((M,q)\) can be extended to an asymmetric normed linear space \((X,q^*)\), i.e., when there exists an asymmetric norm \(q^*\) on \(X\) such that \(q^*|_M=q\). As an application, we study these extensions in the case of subsets of normed lattices.