Solution:

Let each rectangle have shorter side $x$ and longer side $y.$ The outer square has side length $y + x$ and the inner square has side length $y - x.$

Since the area ratio is $4,$ the side ratio is $2,$ so $$y + x = 2(y - x),$$ which gives $y = 3x.$

The ratio of longer to shorter side is $\dfrac{y}{x} = 3.$

Thus, the correct answer is A.