Born in Villa Gobernador Gálvez, Barco began his career with Independiente. He was promoted to the first team and made his debut during the 2016–17 season. Barco played two seasons with Independiente before signing with American side Atlanta United in 2018. At the international level, Barco has represented Argentina at the under-20's.
Barco made his first team – and Primera División – debut on 28 August 2016, coming on as a second-half substitute for Jesús Méndez in a 1–0 away win against Belgrano.[7] He scored his first professional goal on 11 September, netting the last in a 2–0 home defeat of Godoy Cruz.[8]
On 19 January 2018, Barco joined Major League Soccer club Atlanta United as a designated player.[10] The transfer fee paid by Atlanta United was reported to be $15 million, an MLS record.[10] Barco made his debut for Atlanta United on 15 April 2018 in a 2–2 draw against New York City FC, coming on as a 70th-minute substitute for Kevin Kratz.[11] He scored his first goal for the club on 5 May 2018 against the Chicago Fire. He scored the opening goal as Atlanta United won 2–1.[12]
Barco was on the bench for Atlanta United during MLS Cup 2018 against the Portland Timbers.[13] He came on as a substitute in the first minute of second-half stoppage time as Atlanta United won 2–0.[13] Barco then won his second trophy with Atlanta United when the club won 3–2 over América in the Campeones Cup, coming on as a late second-half substitute.[14] A couple of weeks later, on 27 August, Barco was a starter in the U.S. Open Cup final against Minnesota United, helping his side win 2–1.[15]
River Plate
On 30 January 2022, Barco joined Argentine Primera División side River Plate on loan[16] with an option to make the move permanent at the end of 2022 or 2023.[17] On 10 February 2023 it was announced that Barco's loan to River Plate would become permanent on 1 January 2024.[18]
Spartak Moscow
On 25 July 2024, Barco signed a three-year contract with Spartak Moscow in Russia.[19]
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