In mathworld, magic tour, it is mentioned that for odd $n$, only semimagic knight tours are possible on a $n\times\ n$ - board.
For $n = 8$, it has been verified that there are no magic knight tours, only semimagic ones.
What about $6\times6$ and $10\times10$ ? Are there magic knight tours on these boards ?
The case $n = 6$ should be feasible with brute force, but I have no idea how to enumerate the tours efficiently.
There are no magic knight tours on $6 \times 6$ or $10 \times 10$ boards. On this web page from George Jelliss : http://www.mayhematics.com/t/mg.htm, we see "Theorem 4. A magic knight's tour is impossible on a board with singly-even sides." (Here, "singly even" means a number of the form $2k$, where $k$ is odd.)
Now, Jelliss's definition of a magic knight tour differs from MathWorld's definition -- MathWorld requires the diagonals to add up to the same constant as the rows and columns do, while George Jelliss's definition does not. However, any magic tour satisfying the MathWorld definition would satisfy the other definition. So there are no magic knight tours on $6 \times 6$ or $10 \times 10$ boards.
Edit: As noted in the comment below, the answer to "Is there a semimagic tour on any $n \times n$-board, if $n≥5$ ?" is "no".