The 17-year-old, who has been likened to Bayern Munich winger Arjen Robben, had previously been linked with Premier League duo Liverpool and Manchester City.
However, Zivkovic has fended off interest from overseas to continue his development in Holland with Ajax, where he has penned a three-year contract.
"Initially, it was the intention to stay at Groningen for a longer period and become a regular member of the starting XI in the first team," Zivkovic told Groningen's official website.
"But I also have ambitions and, when this chance came, I wanted to take it. But before I will talk about Ajax, I first want to finish the season at FC Groningen in a good way."
Groningen will also be entitled to 10% of any future transfer fee received by Ajax for Zivkovic, who became his club's youngest ever goalscorer in August by netting against NEC Nijmegen.