Solution:

The square has area $10^2 = 100$ and the circle has area $\pi(10)^2 = 100\pi.$

Since the circle is centered at a vertex of the square, exactly one quarter of the circle, area $25\pi,$ lies inside the square.

The union has area $100 + 100\pi - 25\pi = 100 + 75\pi.$

Thus, the correct answer is B.