The 21-year-old, who joined the Eredivisie outfit in 2016, made over a century of appearances for the club and was a member of the squad that claimed a domestic double last season.
A Denmark international, he has also won 13 caps for his country and scored three goals.
He was named Dutch Football Talent of the Year in 2016-17, picking up the award given to the best under-21 player in the Dutch leagues.
Dolberg only started Ajax's Johan Cruyff Shield victory last month but now will trade Northern Europe for the French Riviera on a full-time deal.
Nice had previously only brought in midfielder Khephren Thuram on a free transfer from rivals Monaco this summer, but have been tipped to make some late signings following their takeover by chemicals company INEOS.
45 - Kasper Dolberg a inscrit 45 buts avec l'Ajax toutes compétitions confondues depuis ses débuts avec le club en 2016, plus que tout autre joueur. Aiglon 🦅. pic.twitter.com/ojMUwcjFxX— OptaJean (@OptaJean) August 29, 2019
They previously sold Allan Saint-Maximin to Newcastle United, with several other faces having left during the off-season.
Dolberg passed his medical on Wednesday and arrives having scored more goals for Ajax than any other player since his debut with the club.
He was a member of Denmark's squad at the World Cup in Russia last season, and featured against France.
Patrick Vieira's side, meanwhile, slipped to a 2-1 loss against Marseille on Wednesday but still sit inside the top five in Ligue 1.
Nice are also widely tipped to announce the capture of Lorient star Alexis Claude Maurice in the next few days.
The 21-year-old is thought to have undergone his medical at the same time as Dolberg and his unveiling looks to simply be a formality now.
Linked with moves to Arsenal, West Ham and Norwich City, he is thought to have snubbed the Premier League in order to stay closer to home in France.
Additional reports have also pointed to the capture of Hichem Boudaoui before the transfer window shuts next month.