Bašić joined Hajduk Split's youth academy in 2010, aged 13, after stints at Primorac Stobreč and Omladinac Vranjic.
He was called up to the first team for the game against Dinamo Zagreb on 29 September 2012, starting on the bench, but making his debut by entering for Marko Bencun in the 57th minute, thus becoming the youngest debutant in league history at just 16 years 6 months and 27 days of age.
Bašić scored his first goal for Hajduk's senior team in the national league in November 2013, against Slaven Belupo.
[7] Bašić captained the Croatia national under-17 football team at the 2013 UEFA European Under-17 Championship.
The Croatian U-17 side had quality players such as Alen Halilović, Duje Ćaleta-Car, Ivo Grbić, Robert Murić and Fran Brodić but were ultimately unable to make it out of the group stage finishing 5th at the tournament.