  From the recently known ${\cal N}=2$ supersymmetric linear
$W_{\infty}^{K,K}[\lambda]$ algebra where $K$ is the dimension of fundamental
(or antifundamental) representation of bifundamental $\beta \, \gamma$ and $b
\, c$ ghost system, we determine its ${\cal N}=4$ supersymmetric enhancement at
$K=2$. We construct the ${\cal N}=4$ stress energy tensor, the first ${\cal
N}=4$ multiplet and their operator product expansions (OPEs) in terms of above
bifundamentals. We show that the OPEs between the first ${\cal N}=4$ multiplet
and itself are the same as the corresponding ones in the ${\cal N}=4$ coset
$\frac{SU(N+2)}{SU(N)}$ model under the large $(N,k)$ 't Hooft-like limit with
fixed $\lambda_{co} \equiv \frac{(N+1)}{(k+N+2)}$, up to two central terms. The
two parameters are related to each other $\lambda =\frac{1}{2}\, \lambda_{co}$.
We also provide other OPEs by considering the second, the third and the fourth
${\cal N}=4$ multiplets in the ${\cal N}=4$ supersymmetric linear
$W_{\infty}[\lambda]$ algebra.
