A Fight of Kings!

The white king can be placed on any square among the 64 squares. However the areas it attacks depends on its position.

Case I: If the white king is placed on any other square excluding the edges of the chessboard it attacks 9 squares including the square on which it is placed. Therefore it leaves 64-9 = 55 squares safe. Since there are 64-28 = 36 such squares, therefore there are:

$55 \times 36$ = $1980$ combinations.

Case II: If the white king is placed on the edge of the chessboard (excluding corners) it attacks 6 squares including the square on which it is placed. Therefore it leaves 64-6 = 58 squares safe. Since there are twenty-four such squares of this type, therefore there are:

$58 \times 24$ = $1392$ combinations

Case III : If the white king is placed on any of the four corners of the chessboard it attacks 4 squares including the square on which it is placed. Therefore it leaves 64-4 = 60 squares safe, therefore there are:

$60 \times 4$ = $240$ combinations.

Therefore there are: $1980 + 1392 + 240$ = $\boxed{3612}$ squares where the two kings do not threaten each other.