In July 2014, Calleri signed for Boca Juniors, with coach Carlos Bianchi deciding to take-up the right that the club had with the player, a holding of 30% of the player' rights. On 27 July 2014, Calleri made his Boca debut in a 2–0 loss to Huracan. In the league, his debut came on 18 August as a substitute in a 1–0 win over Belgrano, and a week later coach Rodolfo Arruabarrena started Calleri for the first time in a 3–0 loss to Atlético Rafaela. On 1 November 2015, Boca Juniors were crowned champions of the Argentine Primera División, in which Calleri scored 10 goals in 26 matches and achieved his first title as a professional player. Three days later, Boca Juniors won the 2014–15 Copa Argentina.
Deportivo Maldonado
In January 2016, Calleri joined Uruguayan club Deportivo Maldonado. In an arrangement similar to a number of other players that have been registered to the club, he was loaned to the Brasileirão side São Paulo FC in the same month.
São Paulo (loan)
On 3 February 2016, Calleri made his debut and scored his first goal for São Paulo as they drew 1–1 with Universidad César Vallejo in the 2016 edition of Copa Libertadores. On 13 July 2016, Calleri played his last match for São Paulo in a game they lost 1–2 against Atlético Nacional, in the Libertadores semifinals. He scored the only goal for São Paulo, in the 9th minute. Calleri said that it was his last game for the club and that he had "spent very beautiful six months" playing for Brazilian side. He ended the tournament as the top goalscorer, with 9 goals in 12 matches.
In August, West Ham United signed Calleri on a season-long loan. He made his Premier League debut coming off the bench in the match against Bournemouth on 21 August 2016. His only goal for the club came in the 94th minute in a 3–1 win against Middlesbrough at the Riverside Stadium on 21 January 2017. His full Premier League debut finally came on 22 April 2017 in a 0–0 draw with Everton at the London Stadium. He played 61 minutes before being replaced by Diafra Sakho. After 19 appearances with one goal scored, Calleri's loan ended.
Las Palmas (loan)
On 11 July 2017, Calleri joined La Liga side UD Las Palmas on a one-year loan deal. He made his debut for the club on 18 August, starting in a 0–1 away loss against Valencia CF, and scored his first goal seven days later but in a 1–5 home loss against Atlético Madrid.
Due to some impressive performances with Boca Juniors and São Paulo, Calleri earned a call-up to the Argentina Olympic squad for the 2016 Summer Olympics. On 29 July 2016, Calleri made his Argentina Olympic debut in a friendly against Mexico, which resulted in a 0–0 draw.
Personal life
Calleri is the nephew of Argentine former footballer Néstor Fabbri.