Born in Copenhagen, Tverskov started playing football at the age of three at Boldklubben 1903 (B 1903). As a 17-year-old, he was promoted to the first senior team competing in the fourth-tier Denmark Series by the then manager Bent Christensen. In his first season, he made 24 appearances and seven goals, and was voted Player of the Year at the club.[5][6]
Lyngby
In June 2011, Tverskov moved to Lyngby Boldklub on the recommendation of his manager, Bent Christensen.[5] At that club, he played alongside fellow youngsters such as Yussuf Poulsen, Christian Nørgaard and Uffe Bech.[7] In June 2012, he was promoted to the first-team by head coach Niels Frederiksen.[8] Tverskov made his first professional appearance on 4 November 2012 against Hobro IK in the second-tier Danish 1st Division. He came on as a 90th-minute substitute for David Boysen, as his team won 2–1.[9] He scored his first professional goal on 25 April 2014, during a 3–0 league victory for his team over Hvidovre IF.[10]
Randers
On 30 June 2014, Tverskov joined Randers, where he signed a two-year contract.[11] He played his first match for the club on 23 September in a Danish Cup match against Kolding Boldklub. He started the match as a defensive midfielder and stood out by also scoring his first goal for Randers, thereby participating in his team's 7–1 win.[12] The club finished fourth in the Danish Superliga at the end of the 2014–15 season and managed to qualify for the qualifying rounds of the UEFA Europa League.[13] Tverskov played his first European match in this competition, on 2 July 2015 against Andorran club UE Sant Julià. He appeared as a starter, but was however sent off after receiving a second yellow card, as Randers won 1–0.[14]
OB
As his contract expired with Randers, Tverskov signed with Odense Boldklub (OB) in the summer of 2016, the transfer being announced on 20 April of the same year.[15][16] He made his first appearance for the club on the first matchday of the 2016–17 Superligaen season, against SIlkeborg. That day, he started the match, which ended in a 0–0 draw.[17]
Nordsjælland
On 21 June 2023, Tverskov signed for Nordsjælland, signing a contract until June 2025.[18]
San Diego FC
On 19 March 2024, MLSexpansion teamSan Diego FC announced that Tverskov would join the club ahead of their 2025 inaugural season, signing a two-year contract through 2026, set to be activated on 1 January 2025.[19][20]
^ abHarder, Mads (24 February 2018). "Fra B1903 til Parken". b1903.dk (in Danish). Archived from the original on 30 March 2018. Retrieved 14 March 2021.
^Hoffskov, Ole (30 June 2014). "Randers henter forsvarsspiller i Lyngby". Tipsbladet.dk (in Danish). Archived from the original on 6 March 2019. Retrieved 14 March 2021.{{cite news}}: CS1 maint: bot: original URL status unknown (link)
^"Et stort tab for Randers FC". Din Avis (in Danish). 26 April 2016. Jeppe Tverskov, der med scoring og 7-1-sejr fik debut for Randers FC i en pokalkamp mod Kolding B i september 2014, har været glad for tiden i Randers FC.