Born into a prominent Croatian sports family, Nikola is a youth product of Hajduk Split academy.

[10] On 30 June 2016, Hajduk announced that Vlašić had been appointed vice-captain of the club, with Lovre Kalinić continuing to be the captain.

[11] As Kalinić was on extended holidays as a result of being part of the Croatian squad for the UEFA Euro 2016, Vlašić captained Hajduk for the first time on 14 July, the season opening fixture; a 2–2 away draw against CSM Politehnica Iași in the second round of the 2016–17 Europa League qualifying phase.

On 31 August 2017, Vlašić signed a five-year contract with Premier League club Everton for a fee of around £10 million, which is the Hajduk Split record transfer.

[15] Vlašić scored his first goal for Everton on his debut at Goodison Park on 28 September in a 2–2 draw with Apollon Limassol in the Europa League.

[17] After making only 19 appearances in all competitions, Vlašić was told at end of the season that he was not part of the future plans of new coach Marco Silva.

[23] In the return leg on 12 December, he provided Arnór Sigurðsson with an assist for the third goal in a 3–0 victory, which is Madrid's highest ever European home defeat.

Satisfied with his performances, CSKA announced the signing of Vlašić on a five-year contract for an undisclosed fee on 19 June from Everton.

[26] On 22 September, he won and successfully converted a penalty, provided Fyodor Chalov with an assist and scored in a 3–2 victory over Krasnodar.

[31] At the beginning of the 2020–21 season, Vlašić sparked interest from Napoli, Atalanta and Zenit Saint Petersburg; however, CSKA refused to sell the player.

[38] On 31 August 2021, Vlašić joined Premier League club West Ham United on a five-year contract for an undisclosed fee.

[47] On 11 August 2022, Vlašić joined Torino on-loan for the remainder of the 2022–23 season with an option to buy the player at the end of the loan.

[51] On 18 November 2018, he provided Andrej Kramarić with an assist in a UEFA Nations League match against England for a 0–1 lead, which Croatia eventually lost 2–1.

[57] On 11 October 2020, Vlašić scored the opener in a 2–1 UEFA Nations League victory over Sweden at Stadion Maksimir and followed it up three days later with a goal in a 2–1 loss to reigning world champions France at the same location.