The Dutch, who also failed to reach Euro 2016 in France last year, needed a seven-goal win against Sweden in their final Group A game on Tuesday night.
And, while Arjen Robben's brace ensured they did win the game 2-0, it is the Swedes who will go into the draw for next month's qualification play-off after finishing second behind group winners France.
Dick Advocaat is set to remain in charge for some upcoming friendlies and the former Rangers and Sunderland boss is then likely to make way for a successor ahead of the Euro 2020 qualifiers, which will get under way in 12 months' time.
Koeman was linked with the job during his time at Southampton and it seems he is one of the favourites to replace Advocaat as he has the backing of Dutch newspaper De Telegraaf, with whom he is a regular columnist.
It is not known how Everton bosses would react if an approach was made for Koeman, but as he is already under pressure given that the Toffees currently sit in 16th spot in the table after a huge outlay on transfers in the summer, he could potentially be a free agent by the time the Netherlands come calling.