Born in Los Angeles, California, Tarek moved to Egypt at early age, settling in Asyut. After being coached by a German manager at Asyut Cement, he joined ENPPI Club in 2009, aged 17; at that time he was mainly used as a striker. In May 2010 Tarek moved to Europe, joining German side FC Magdeburg. Initially on a trial basis, he was subsequently offered a contract and was assigned to the under-19 squad. In the 2011 summer Tarek joined SC Freiburg, being assigned to the reserves in Regionalliga. He scored the first in a 6–0 victory over Serbian side OFK Beograd in the 2011 Lev Yashin Cup final, also featuring regularly during the campaign; he was also converted initially to a left winger and later to a left back in the process. On 3 May 2012 Tarek signed for another reserve team, VfL Wolfsburg II. Despite extending his contract with the club roughly a year later, he moved to HNK Hajduk Split in August on a trial basis, and after nothing came of it, he returned to the United Statesin September and played 82 minutes in a 4–1 MLS Reserve League win for Chivas USA Reserves against Portland Timbers Reserves.
On 6 July 2015 Tarek signed a four-year deal with Real Betis, newly promoted to La Liga, for an undisclosed fee. On 13 June 2016 Real Betis announced the termination of Amro Tarek’s contract by mutual consent.
On 2 February 2016, Betis announced that Tarek had joined Columbus Crew of Major League Soccer on loan until June 2016 with the option to extend the loan until December 2016. Tarek struggled for playing time with Columbus, and was released by the club on May 5, 2016 after just a single substitute appearance.
On 15 June 2016, just two days after his contract termination with Real Betis, Tarek returned to Egypt and signed a four-year deal with Ismaily. However, the deal was terminated a month later before his registration was submitted, with Tarek joined fellow Egyptian Premier League side ENPPI. After one season with ENPPI, Tarek terminated his deal.
In February 2018 Tarek made his second loan move to MLS, this time joining Orlando City. He made his first start in the season opener versus D.C. United and scored his first goal for the club in a 3–3 draw against New England Revolution.
On 11 December 2018, it was announced that Tarek had been traded to New York Red Bulls in exchange for a fourth-round pick in the 2019 MLS SuperDraft. On 2 March 2019, Tarek made his official debut with the club, appearing as a starter in a 1-1 draw with Columbus Crew on the opening day of the season. On 4 May 2019, Tarek opened the scoring for New York in a 3-2 victory over LA Galaxy. At the conclusion of the 2019 season he was named as the teams Newcomer of the Year.
