Koeman stepped down immediately after the Oranje's exit from the World Cup, where they fell to Morocco on penalties in the round of 32 following a stoppage-time equaliser that forced extra time, having taken over from Louis van Gaal following the Qatar tournament.
Announcing his decision, the 63-year-old said the time felt right to prioritise his family.
"I have decided that it is now the right time to spend more time with my wife, children and grandchildren," he said.
Koeman's resignation immediately sparked speculation over who might take on the role before the Netherlands' next major tournament, with Slot's name having first emerged in reports last week.
Koeman's exit brought an end to his second spell as Netherlands boss, having originally managed the side between 2018 and 2020, before departing for Barcelona, then returning to the post in 2023 following Louis van Gaal's retirement.
Slot left Liverpool at the end of the season, bringing to a close a two-year spell at Anfield that included a Premier League title in his first campaign in charge.
Sky Sports reports that the Dutch federation has now opened preliminary talks with the 47-year-old over the vacancy, though he is said to be just one of several candidates under consideration. The identities of the other names being considered have not been made public, and it remains to be seen whether these early discussions progress into formal negotiations.
Slot previously managed Feyenoord and AZ Alkmaar before moving to the Premier League, building a reputation as one of Dutch football's most promising young coaches long before he arrived in England, where he was seen as the natural successor to Jurgen Klopp's attacking style.
Liverpool finished fifth in what proved to be his final season in charge, a disappointing conclusion to his time at the club following the title success that preceded it.