This paper shows how to create magic squares with a perfect square number for the total sum of their entries. This has been done in two ways: Firstly, by using the sum of consecutive odd numbers, and secondly, by using consecutive natural numbers. In the first case, for all orders of magic squares, one can always obtain a perfect square entries sum. In the second case, magic squares with perfect square magic sums do exist, but only for odd order magic squares. For the even order magic squares, such as, 4, 6, 8, etc. it is not possible to write consecutive natural number magic squares with perfect square entries sums. A simplified idea is introduced to check when it is possible to obtain minimum perfect square entries sums. Also, a uniform method is presented so that, if k is the order of a magic square, then the magic sum of the square is k^3, and the sum of all entries of the magic square is k^4. Based on these aspects, connections with Pythagorean triples are also made. The work is for the magic squares of orders 3 to 31. Further orders shall be dealt later on. In another work, the magic squares are generated based on Pythagorean triples.