Osmokrović, then Nataša Leto, played with her national team in the 2000 Summer Olympics, helping them to reach the quarterfinals rounds and finishing in seventh place. She won the bronze medal at the 2007–08 CEV Indesit Champions League with Asystel Novara and also was individually awarded "Best Receiver". She also won the 2009 CEV Cup with Novara. She signed a two-year contract with the Turkish team Fenerbahçe Acibadem in 2009. With this team she was awarded "Best Server" at the CEV Indesit Champions League Final Four 2010 and win the Turkish League Championship. She also won the 2010 Clubs World Championship crown and the bronze medal in the 2011 CEV Championsleague with Fenerbahçe.