On 25 June 2019, Drmić joined Norwich City on a free transfer, signing a three-year deal. He was persuaded to join them by their Swiss defender Timm Klose. He made his Premier League debut on 8 August, coming on as an 83rd-minute substitute for Teemu Pukki in a 4–1 loss at European champions Liverpool. On 5 October, again as a late replacement for the Finn, he scored his first goal as consolation in a 5–1 home loss to Aston Villa.