Solution:

Let $r_1$ and $r_2$ be the roots of $x^2 + px + m = 0,$ so $m = r_1 r_2$ and $p = -(r_1 + r_2).$

The roots of $x^2 + mx + n = 0$ are $2r_1$ and $2r_2,$ so $n = 4r_1 r_2$ and $-m = 2(r_1 + r_2).$

Then $n = 4m$ and $m = -2(r_1 + r_2) = 2p,$ so $p = \dfrac{m}{2}.$ Therefore $$\dfrac{n}{p} = \dfrac{4m}{m/2} = 8.$$

Thus, D is the correct answer.