A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $λK_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $μK_{m}$ for some range of values for the parameters $n$, $m$, $λ$, $μ$, $r$: $r=2$, $μ>λ$ and either $m \geq 2n-1$, or $m=2n-2$ and $μ= 2$ and $λ=1$, or $n=3$ and $m=4$. We generalize their result to every $r \geq 2$ and $m \geq 2n - 2$. We also give some sufficient conditions for enclosing a given decomposition of $λK_n$ in some $2$-edge-connected $r$-factorization of $μK_{m}$ for every $r \geq 3$ and $m = (2 - C)n$, where $C$ is a constant that depends only on $r$, $λ$ and~$μ$.