Field at the center of smaller square loop due to current in the larger loop $ B = 2\sqrt 2 \frac{\mu_0 I}{\pi L}$ along the axis.

Flux passing through the smaller loop is $\phi = B l^2 = 2\sqrt 2 \frac{\mu_0 I}{\pi L} l^2$

$ \text{Mutual Inductance is } M = \frac{\phi}{I} = 2\sqrt 2 \frac{\mu_0 l^2}{\pi L}  $