Born in Zarautz, Basque Country to an Algerian father and a Spanish mother, Berchiche started his career in the cantera of Real Sociedad, but left for rivals Athletic Bilbao at the age of 16. On 8 June 2007 Athletic and Tottenham Hotspur agreed on a transfer, and the player continued his footballing development in England at the club's academy; while playing for Tottenham's under-18 at the Eurofoot Tournament in Belgium, he was named "Player of the tournament", as Spurs won the competition after beating R.S.C. Anderlecht 4–0 in the final. Berchiche joined League One team Cheltenham Town on a month's loan, on 26 March 2009. Two days later he made his league debut, in a 1–1 away draw against Walsall. In April 2009, Berchiche made a robust tackle on Scunthorpe United's Henri Lansbury, leaving his opponent with a suspected broken leg. Although he was not sent off for the challenge, Cheltenham manager Martin Allen ordered him to leave the field anyway; it was later revealed that Lansbury did not sustain any major damage to his leg. In July 2009, Berchiche returned to Spain and joined Real Valladolid, still on loan. Benefitting from the absence of longtime incumbent Alberto Marcos due to injury, he started in the first league game, a 0–0 draw at UD Almería; it would be his only La Liga appearance of the season as he was mainly registered by the reserves, and the Castile and León side also suffered relegation.
Real Sociedad
In the summer of 2010, aged 20, Berchiche was released by Tottenham, signing a two-year contract with Real Unión in the Spanish third division. On 29 June 2012, after two full campaigns, he returned to Real Sociedad, who immediately loaned him to another team in the third level, neighbouring SD Eibar. In early 2014, Berchiche renewed his contract with the Txuriurdin until 2016. In June, after helping Eibar obtain promotion to the top divisionfor the first time in its history, he was selected by the Liga de Fútbol Profesional to its starting XI. In August 2014, Berchiche returned to Real Sociedad, being assigned to the main squad. He subsequently became the starter, replacing longtime incumbent Alberto de la Bella.
On 2 July 2018, PSG announced they had agreed a deal with Athletic Bilbao for the permanent transfer of Berchiche. The contract was announced as running through 2022, with a release clause of €100 million.