Does this number even exist?

Let the number be $\overline {aabb}$

We know $100^2=10000$ , Hence, The number has to be a two digit number

From the divisibility rules of 11, we get to know that this number is divisible by 11.

We also know from the given criteria, That the number is a perfect square.

Hence, The number has to be in the form of:- $11^2 \times a^2$ Where $0>a>=9$

We list out all such numbers.

$121 \times 1=121$

$121 \times 4=484$

$121 \times 9=1089$

$121 \times 16=1936$

$121 \times 25=3025$

$121 \times 36=4356$

$121 \times 49=5929$

$121 \times 64=7744$ < Satisfies the Criteria

$121 \times 81=9801$

Hence, The only number which satisfies the conditions is 7744.

Answer:- $\boxed {7744}$