When he was seven months old, his father, Nestor Cortés Sr., won the visa lottery, and they relocated to the United States in the city of Hialeah, Florida.

[2] His father took him to see the Florida Marlins of Major League Baseball (MLB) play at Pro Player Stadium in Miami Gardens.

Cortés spent 2015 with the Pulaski Yankees of the Rookie-level Appalachian League, where he compiled a 6–3 record with a 2.26 ERA in 12 games (ten starts).

[17] After the regular season, Cortés played for the Estrellas Orientales of the Dominican Professional Baseball League, where he learned how to throw a cut fastball.

[23] On November 25, 2019, the Yankees traded Cortés to the Seattle Mariners in exchange for $28,300 of international bonus pool money.

[27] On December 20, 2020, Cortés signed a minor league contract to return to the New York Yankees organization.

His signature mustache, along with his unique style of pitching and many productive appearances, garnered him much praise, and brought the moniker "Nasty Nestor.

[34] Working exclusively as a starter, he finished the regular season with a 12-4 record and a 2.44 ERA in 158+1⁄3 innings pitched in 28 games.

He struggled, posting a 5.16 ERA across 11 starts, before being placed on the injured list with a left rotator cuff strain on June 8.

[41] He was transferred to the 60–day injured list on September 12, officially ending his season, as the Yankees missed the postseason for the first time since 2016.

[42] With Gerrit Cole beginning the 2024 season on the injured list, the Yankees chose Cortes as their starting pitcher for Opening Day.

Cortes was sidelined to begin the Yankees' 2024 postseason run due to a left elbow flexor strain suffered on September 18.

In the bottom of the 10th inning of Game 1, Cortes retired the first batter he faced, then allowed a walk-off grand slam to Freddie Freeman.

[44] On December 13, 2024, the Yankees traded Cortés and Caleb Durbin to the Milwaukee Brewers in exchange for Devin Williams.

[6] Cortés hides the ball well in his wind-up, which helps to alleviate the problems presented by his shorter frame.