A product of Hajduk Split academy, Radošević had several loan spells between 2007 and 2010 at lower level Croatian sides and at Željezničar in Bosnia and Herzegovina, before having his league debut for Hajduk in the 2010–11 season on 24 July 2010 against Istra 1961. In the 2010-2011 season Radošević played in Europa League for Hajduk Split. Radošević debuted in qualifying match against Unirea and also performed in the group stage of competition in matches against AEK and Anderlecht. Radošević was also capped for Croatia at youth levels, appearing in 15 matches for under-17 and under-19 national teams in 2005 and 2006. In the 2012 Radošević was the most responsible player for the remaining Solin among Croatian Second League, which brought him a transfer to Istra 1961. In Istra 1961 Radošević be kept only 8 months and in March 2013 terminated the contract because of debts and disagreements with coach Igor Pamić. In March 2013 temporarily joins the team Šibenik, where he appeared in 11 games and in June 2013 becomes a free agent. In July 2013 he signed a contract with Inter Zaprešić. After a great season in Inter Zaprešić, Radošević goes to Slovenia in Koper. Radošević is most remembered for his match against Neftchi Baku, which is the best result for Koper in Europa League. During the season the club was facing the result crisis after the best players left the club. This prompted president Ante Guberac to look for culprits in the players he personally led. From the first team are removing all foreign players led by Radošević. Players were faced with extremely inhumane working conditions and threats by the president Ante Guberac. After a month of work in inhumane conditions, some players have to agree with the club. Radošević has left the club in November 2014 and became a free agent. In January 2015 Radošević signed half year contract with Hungarian second division club Balmazújváros with the mission of saving the club from relegation. Radošević and another five Croatian players have made a big miracle and they ensure the second division status for Balmazújváros. His great form in Hungary have not gone unnoticed by other clubs and in June 2015 signed two years contract with one of the biggest clubs in Hungary Debreceni VSC. He made his debut against Puskás Akadémia in a draw 1-1. After qualifying matches in Europa League and the failure of Debreceni VSC, he accepted the offer from Iranian club Persepolis and signed contract for two years. In his new club, he made debut against Sanat Naft in front of 80,000 Persepolis fans. Persepolis won the game and Radosevic provided a clean sheet during the 90 minutes. Radošević enjoys great popularity in Iran and Tehran. He is with his loyalty, work and behavior deserved a special place in hearts of all Persepolis fans forever. Persepolis became the champion after 9 years with impressive statistics in Persian Gulf Pro League and also very successfully performed in the AFC Champions League 2017 where they have reached the semifinal. His second season didn’t perform that much, only 3 games but Radošević won Super Cup and again Persian Gulf Pro League. The team also reached the finals of the AFC Champions League 2018 which they lost against Kashima Antlers. On 19 April2018, he signed a three years contract extension with the Tehran-based club. Radošević third season in Persepolis was also successful. The team won three competitions Persian Gulf Pro League, Hazfi Cup and Super Cup. Radošević performed well in some matches but still was second choice in Persepolis. The first goalkeeper was Alireza Beiranvand. At the end of the season Croatian coach Branko Ivanković left the team and signed contract with Saudi Professional League team Al-Ahli. After Ivanković era Argentine coach Gabriel Calderon becomes the new coach of Persepolis. After three years of waiting, Radošević gets a chance as the first goalkeeper in front of Beiranvand which was an unexpected move by Calderon. First half season Rado performed perfectly without conceding a goal. He was selected for the half season best eleven in the Persian Gulf Pro League. Calderon left the bench in December 2019. In February 2020 he signs a new four-year contract making him the longest-serving foreigner in Persepolis history.
