Esposito made his professional debut for Inter Milan at 16 years old, on 14 March 2019, in the second leg of Europa League match against Eintracht Frankfurt, coming in as a substitute for Borja Valero in the 73rd minute.

The following season, on 23 October 2019 he made his debut in Champions League, replacing Romelu Lukaku in the group stage match against Borussia Dortmund, and becoming the first player born in 2002 and the second youngest ever for the club to feature in a Champions League/European Cup match, after Giuseppe Bergomi.

[1][2][3] On 26 October, he made his Serie A debut, aged 17, coming as a substitute for Lautaro Martínez in a home match against Parma in San Siro.

On 21 December 2019, Esposito scored his first goal for Inter, on his full debut, from a penalty in a 4–0 win at home against Genoa.

[8] Three days later, on 25 July, Esposito made his Swiss Super League debut for Basel against Grasshoppers and scored his first goal for the club in the same game as they recorded 2–0 victory.