Born in Santander, Cantabria, Concha joined Racing de Santander's youth setup at the age of 9 following a brief stint with lowly CD Marina Sport.[1] He was called up to train with the main squad in December 2013, being also selected for the Copa del Rey match against Sevilla FC.[2] He made his first-team debut on the 6th, coming on as a substitute for Javi Barrio in the 0–1 home loss.[3] He first appeared in the league four days later, again coming from the bench in a 1–0 home win over UD Logroñés in the Segunda División B.[4]
On 8 January 2014, Concha scored his first goal as a senior, helping to a 1–1 home draw against UD Almería also in the domestic cup.[5] He finished the season with 13 appearances, with Racing returning to Segunda División at the first attempt.
Concha played his first match as a professional on 24 August 2014, starting in a 1–0 defeat at Girona FC.[6] He scored his first goal in the division on 21 September, opening an eventual 2–2 home draw with CD Leganés.[7]
Concha made his top-flight debut on 21 August 2016, starting in a 0–3 home loss to Real Madrid.[10] The following 11 August, he was loaned to FC Barcelona B in the second division for one year,[11] but his loan was cut short on 31 January 2018.[12]
On 9 March 2019, after more than one year of inactivity, Concha moved abroad for the first time in his career after agreeing to a one-year loan with Gamba Osaka.[13] On 27 February 2020, after only two official games, he returned to Spain and joined to CD Badajoz on a short-term contract.[14]
On 6 July 2022, Concha signed a two-and-a-half-year contract with Swedish Allsvenskan side Hammarby Fotboll.[15] The following January, having made 14 competitive appearances, he left by mutual consent and announced his retirement,[16] but returned to active on 11 July 2023 on a one-year deal at Primera Federación's Gimnàstic de Tarragona.[17]
^"David Concha lämnar Hammarby" [David Concha leaves Hammarby] (in Swedish). Hammarby Fotboll. 25 January 2023. Archived from the original on 24 September 2023. Retrieved 25 January 2023.
^"David Concha signa pel Nàstic!" [David Concha signs for Nàstic!] (in Catalan). Gimnàstic Tarragona. 11 July 2023. Retrieved 11 July 2023.