The linking number is the same as the homology class that one component represents in the complement of the other. You can reduce any $2$-component link to a normal form by first homotoping one component to be an unknot. Then the complement of this component is homeomorphic to a solid torus, which has $\pi_1\cong\mathbb Z$. So all loops can be homotoped to one representative for each $n\in\mathbb Z$, where $n$ is the linking number.