Radošević started his career playing at youth level for his hometown club Hajduk Split. The talented midfielder was promoted to Hajduk's senior side in November 2011. On 23 November 2011, he made his debut for the first team in a 1–0 win over NK Zagreb in the quarter-finals of the 2011–12 Croatian Cup, aged 17 and 7 months. In the 2011–12 season, Radošević started featuring as a regular in Hajduk first eleven, after Mišo Krstičević, who had previously coached Radošević in Hajduk Split youth team, was appointed as the new manager. In the 2012–13 season he established himself as the key player in defensive tasks in Hajduk midfield, appearing in 17 domestic league games before being transferred to S.S.C. Napoli.
Napoli
On 29 January 2013, Radošević was loaned to Serie A side Napoli for a fee of €2 million euros. He will be on loan until this summer when Croatia becomes a part of the EU so he won't be a foreign player when he signs his contract in the summer. Hajduk Split will get another €1 million euros when he completes his transfer. On 25 August 2013, he made his official debut for Napoli in the first round of the 2013–14 season against Bologna, coming on as a substitute for Gonzalo Higuaín in the 84th minute.
On 4 February 2015, Radošević was loaned to Rijeka until 31 December 2015. He scored his first goal for the club in a dramatic fashion, when he scored an equaliser in the fourth minute of stoppage time against Dinamo Zagreb on 4 April 2015.
Josip Radošević was announced as a new Hajduk player on 3 June 2017. He took the kit number 14, the same one he had four and a half years ago before he left for Napoli.
Brøndby
On 27 July 2018, it was announced that Radošević would join the Danish side Brøndby IF on a four-year contract for a fee of €800,000. He made his debut on 5 August in a 2–0 Danish Superliga win over FC Nordsjælland. Radošević made 37 appearances in his first season in Denmark, scoring two goals. During his first season he mostly appeared as a defensive midfielder, taking over the role from fan favourite Christian Nørgaard who had signed with Fiorentina before the start of the season. In the following season, after Niels Frederiksen took over as head coach of Brøndby, Radošević continued as a starter on defensive midfield before suffering a shoulder injury in a 5-2 win over Randers FC on 27 October 2019.
International career
Radošević was a part of the Croatian national under-17 team in the qualifications for the 2011 UEFA European Under-17 Football Championship, playing in all six group games. Although Croatia ended qualifications undefeated, in the elite round they finished runners-up in the group behind hosts Netherlands and failed to qualify. Overall, he gathered 35 appearances for the youth teams, scoring two goals. In August 2012, Radošević was called up to the national team by manager Igor Štimac for the 2014 FIFA World Cup qualification matches against Macedonia and Belgium as an alternative for the defensive midfielder Ognjen Vukojević. On 11 September 2012, Radošević made his full debut for the national side in a 1–1 draw against Belgium, making him the youngest player to debut in the history of the national team.
