The 21-year-old, scorer of 17 goals this season for Nuremberg, has agreed a contract through to 2019 with Leverkusen, who finished fourth in Germany's top-tier Bundesliga and will take part in the Champions League playoffs next season.
According to SID, an AFP subsidiary, Leverkusen will pay a transfer fee in the region of seven million euros ($9.6million) for Drmic, a Swiss footballer of Croatian descent.
"Josip Drmic showed his exceptional quality over the last Bundesliga season," Leverkusen chief Michael Schade told the club's website.
"He is one of the best goal-scorers in our championship and with him we want to improve our attacking threat and strengthen the team for another Champions League campaign."
Drmic's goals could not stop Nuremberg from being automatically relegated to the second tier of the Bundesliga after a 4-1 defeat at Schalke on Saturday.
The striker began his career in 2012 with FC Zurich of the Swiss Super League before joining Nuremberg last year.