Souquet was born in Paris and began his career playing for CO Vincennois, the same club that produced the likes of Blaise Matuidi and Yacine Brahimi. While attending the Vincennes-based club, Souquet had the opportunity to join both the famous Clairefontaine academy and the Parisian giants Paris Saint-Germain. However, he declined the chance and after his father met with then-president Alain Cayzac of the latter, he declined that chance, as well.
Souquet joined Lille in 2007 as a youth player after the club spotted him while playing with Vincennes. After excelling in the club's youth system and internationally, for the 2008–09 season, he was promoted to the club's Championnat de France amateur team, despite being only 16 years old at the time. He made his debut with the reserves in their 1–5 defeat to AS Nancy reserves on 14 September 2008. Over the course of the season, he made 12 appearances, which included nine starts. Following the season, on 17 July 2009, Souquet signed his first professional contract agreeing to a three-year deal. After starting the season with the club's Championnat de France amateur team, on 30 September 2009, due to injuries to midfielders Florent Balmont and Ludovic Obraniak, manager Rudi Garcia announced that Souquet would be included in the team for the club's match against Czech club Slavia Prague in the UEFA Europa League. Souquet was assigned the number 35 shirt and made his professional debut, as a starter, playing the entire 90 minutes. In the match, which Lille won 5–1, Souquet scored his first professional goal in the 88th minute and also assisted on a goal scored by Pierre-Alain Frau. On 17 October, he made his league debut starting in the club's match against Rennes. The match ended 0–0 with Souquet only playing the first half.
Souquet has played with both the under-16s and the under-17 team. He was a regular in the under-17s during the run up to the 2009 UEFA European Under-17 Football Championship. During the qualification process, he scored goals against Scotland and Denmark. The 1–0 victory over Denmark assured progression to the tournament, where France were eliminated in the group stages. In total with the under-17s, Souquet made 15 appearances and scored two goals.
