Solution:

Adding the numbers in the top row shows that every row and column must have sum $12$.

In the first column, the missing number above $x$ is $14-x$. In the bottom row, the missing number to the right of $x$ is $4-x$. In the middle row, the remaining missing number is $x-1$.

Since $x$ is larger than the other missing numbers, we need x>14-x, x>4-x, and $x> x-1$. The strongest condition is x>7, so the smallest possible integer value is $8$.

Thus, the correct answer is D.