Born in Rosario, Santa Fe Province, Correa joined San Lorenzo’s youth setup in 2007, aged 12, after a trial period. In the summer of 2012, he was due to join Portuguese club Benfica on a free transfer, but the deal later collapsed.

Correa signed a four-year professional deal with the Ciclón on 23 September 2012, and was promoted to the first-team in January 2013. On 31 March he played his first match as a professional, coming on as a second-half substitute in a 0–1 loss at Newell’s Old Boys.

On 27 May 2014, after scoring six goals in 2013–14, Correa had agreed a deal with La Liga holders Atlético Madrid for a reported fee of €7.5 million. He signed a five-year contract with the Madrid side, joining the club after the 2014 Copa Libertadores.

In June 2014, however, Correa was sidelined for six months due to a heart tumour. After undergoing surgery on 18 June in New York, he started a light training in August, and officially joined the club on 13 December.

On 22 August 2015, Correa finally made his Atleti debut, replacing Óliver Torres in a 1–0 home league match win against UD Las Palmas. He scored his first goal for the club on 19 September, netting the first in a 2–0 away win against SD Eibar; he also assisted Fernando Torres in the second goal.

On 6 November 2019, Correa marked his 200th appearance for Atlético, starting in a 1–2 away loss against Bayer Leverkusen. On 10 August, Atlético announced that Correa and teammate Šime Vrsaljko were both positive for COVID-19, ruling them out of the team’s quarter-final fixture against RB Leipzig.