Career
FK Dukla Banská Bystrica
Born in Revúca, Gyömbér started playing football with club MFK Revúca, before moving to the youth team of Corgoň Liga club FK Dukla Banská Bystrica in 2006. Norbert made his professional debut with Dukla Banská Bystrica on 1 October 2011 in a league match against MFK Ružomberok. On 25 May 2012, Gyömbér was voted by Dukla Banská Bystrica"s fans as player of the season, despite starting in just 18 league matches.
He was also named Revelation of the 2011-2012 Corgoň Liga"s season in July 2012, after his successful first season with FK Dukla Banská Bystrica.
Calcio Catania
Calcio Catania officially agreed to the permanent transfer of midfielder Norbert Gyömbér from Dukla Banská Bystrica during the 2013 January transfer window, though the transfer was not officially be completed until 1 July 2013. The 20-year-old agreed a four-and-a-half-year deal at the Stadio Angelo Massimino, but finished the 2012-2013 season with the top-flight Slovakian side.
Roma
On 18 August 2015, he joined Roma on a one-year loan.