Born in Montevideo, Varela started his career with his hometown club Peñarol and made his senior debut on 5 June 2011, in a 1–0 home defeat against Racing Club de Montevideo, on the last matchday of Uruguayan Primera División.

In May 2013, Varela went on a two-week trial with Manchester United after an impressive performance at the 2013 South American Youth Championship.

On 7 June 2013, Manchester United announced they had reached an agreement to sign Varela from Peñarol for an undisclosed fee, rumoured to be €2.8 million. On 11 June 2013, he signed a five-year contract with United. Hence, he became the first player to join Manchester United under new manager David Moyes.

In September 2014, Varela joined Real Madrid reserve side Real Madrid Castilla for season-long loan. He was, however, included in the 2014–15 UEFA Champions League squad for the first-team and he was given the squad number 28. He made 33 appearances and scored once, when he was set up by Martin Ødegaard in a 4–0 win over Barakaldo on 21 February 2015, at the Estadio Alfredo Di Stéfano in the season’s Segunda División B.

After returning to Manchester United, Varela was included in Manchester United’s squad for the group stage of the 2015–16 UEFA Champions League. He made his debut on 5 December 2015, as a half-time substitute for Paddy McNair in a goalless Premier League draw against West Ham United at Old Trafford. Three days later, Varela made his first start and Champions League debut in the 2015–16 UEFA Champions League, playing the full 90 minutes in a 3–2 defeat to Wolfsburg. On 25 February 2016, in a Europa League Round of 32 return leg against FC Midtjylland, he assisted on the second of the two goals Marcus Rashford scored on his Manchester United debut. On 19 April 2016, Varela scored an injury-time winner to win Manchester United Under-21s the 2015–16 U21 Premier League title.

After the arrival of José Mourinho as Manchester United manager in 2016, Varela came on as a substitute in Mourinho’s first match in charge of the team (a 2–0 win away to Wigan Athletic in a friendly). However, he was not included in Mourinho’s squad for the club’s summer tour of China, and on 23 July 2016, Varela joined German club Eintracht Frankfurt on a season-long loan deal.

On 5 March 2016, following his emergence into the Manchester United first-team, Varela received his first call-up to the Uruguay senior squad, for the World Cup qualifiers against Brazil and Peru later that same month.

Frankfurt coach Niko Kovač revealed at the end of the season that Varela was expected to play in the 2017 DFB-Pokal Final and be given an extension, but these plans were abandoned after Varela defied the instructions of the club’s training staff and got a tattoo, which later became inflamed. Because of this, and other disciplinary issues, Varela was suspended by the club, who said that they would not be attempting to extend his loan deal.

On 12 August 2017, Varela returned to Peñarol on a permanent deal. He made his second debut for the club on 20 August 2017, in 4–0 win against El Tanque Sisley.

On 20 December 2018, Varela joined Danish Superliga team FC Copenhagen on a deal until June 2023.

In June 2018, Varela was named in Uruguay’s final 23-man squad for the 2018 FIFA World Cup in Russia.

On 17 October 2020, he joined Russian Premier League club FC Dynamo Moscow on loan for the 2020–21 season.