[2] In late 2019 Véliz had his first trials with Rosario Central,[2][3] but he only started playing for the club more than a year later as youth competitions had been halted by the COVID-19 pandemic.

[4] Véliz made his professional debut for Rosario Central on 23 July 2021, replacing Alan Marinelli during a 1–0 home victory against Venezuelan side Deportivo Táchira in the round of 16 of the Copa Sudamericana.

[2][3] He played only a small number of games in the Primera División that season but it was during the next that he made a breakthrough, scoring his first goal on 2 May 2022 during the 2–1 away Copa de la Liga loss to Club Atlético Huracán.

[3][5][6] On 21 July 2022, Véliz scored the only goal of Rosario Central's Primera División victory against Newell's Old Boys in the historic Clásico Rosarino.

[11][12] On 8 August 2023, Véliz signed a six-year deal with English Premier League club Tottenham Hotspur in a transfer estimated to be worth £13 million.