If square of a $3$-digit number ends with itself, then the square of its last two digits must end with this two digit number as well. There are only two such numbers with this property: $25$ and $76$. Therefore this three number must end with $25$ or $76$

Only one $3$-digit number which ends in $25$ whose square ends with itself: $625$.

One $3$-digit number which ends in $76$ whose square ends with itself: $376$.

Therefore, the sum of these two numbers is $625+376=\boxed{1001}$.