1. **State the problem:** We have a square loop of side length $l$ with a circular region of magnetic field of radius $R$ inside it. The magnetic field $B_0$ is directed into the plane and exists only within the circle. 2. **Understand magnetic flux:** Magnetic flux $\Phi$ through a surface is given by the integral of the magnetic field over the area it penetrates: $$\Phi = \int B \cdot dA$$ Since $B_0$ is uniform and directed into the plane only inside the circle, the flux linked with the loop is the magnetic field times the area of the circle inside the loop. 3. **Calculate the flux:** The area of the circular region is: $$A = \pi R^2$$ Since the magnetic field is uniform and only inside this circle, the flux is: $$\Phi = B_0 \times \pi R^2$$ 4. **Check if the circle fits inside the square:** The problem implies the circle is inside the square loop, so the entire circular area contributes to the flux. **Final answer:** $$\boxed{B_0 \pi R^2}$$ This corresponds to option A.