Just a thought if anyone doubts that the impetus to leave is coming from RK rather than the club, consider this. If RK doesn't want to sign a new deal and the board says that he has to or will be sacked, then he can call their bluff. If the board sack him, then he is free to join whoever he likes and since there would be no compensation to pay there would be more money available for his salary. Obviously, the board wouldn't want a manager who wasn't committed to the job but RK just has to sit tight. It's obviously him who has pushed for this move.
But if the board went and said something along those lines, not as blatant as that and then Koeman's agent tells him we can sort this, then talks to Everton, then his agent messed up the chance to sit tight and therefore all the cards then lie with the Saints.
I think that is a possibility.