Born in Berlin to Bosnian parents, Bešić started playing football at local clubs, before joining Hamburger SV’s youth academy in 2009. He made his professional debut against Borussia Dortmund on 12 November 2010 at the age of 18.

In November 2010, he received his first senior call-up, for friendly game against Slovakia, and debuted in that game on 17 November. Bešić thereby became the youngest player to ever play for Bosnia and Herzegovina on senior level. The record was previously held by Miralem Pjanić.

In March 2012, Bešić was suspended from the first team, due to lack of discipline. Bosnian media also claimed that manager Thorsten Fink “strangled” Bešić and threw him out of the dressing room.

In August 2012, Bešić moved to Hungarian team Ferencváros on free transfer. On 5 May 2013, he scored his first professional goal against Paks.

He is a practicing Muslim; along with international teammates Ibrahim Šehić, Armin Hodžić, Izet Hajrović, Sead Kolašinac, Edin Višća and Ervin Zukanović he visited a mosque in Zenica during national team concentration. On 4 July 2013, he became the Bosnian ambassador for the Emmaus charity organisation.

On 24 July 2014, Bešić was transferred to English side Everton for an undisclosed fee. Transfer fee was speculated to be in the region of €5 million.

In June 2014, Bešić was named in Bosnia and Herzegovina’s squad for 2014 FIFA World Cup, country’s first major competition. He made his competition debut in the opening group match on 15 June, which ended in narrow loss to Argentina at the Maracanã Stadium.

He was described in 2014 by Everton manager Roberto Martínez as a “very complete footballer”, who “has got a very natural balance about knowing that he can be strong and aggressive but in the same way he’s a technical player on the ball.”

In July 2015, Bešić picked up a hamstring injury in pre-season game against Swindon Town, which ruled him out for the start of season. He made his season debut in EFL Cup game against Barnsley on 26 August.

In February 2016, Bešić was voted Everton’s Player of the Month for January.

On 29 February 2016, he prolonged his contract until June 2021.

In July 2016, Bešić changed his shirt number from 17 he wore for previous two seasons to 21.

In late December, he began with light training, and he was given medical all-clear in March 2017.

Bešić returned to pitch in UEFA Europa League play-offs against Hajduk Split on 17 August 2017, over a year since the injury.

In January 2018, Bešić was loaned to Championship outfit Middlesbrough until the end of season. He debuted for the team on 17 February against Cardiff City. Two months later, in a victory over Derby County, he scored his first goal for Middlesbrough.

In August 2018, his loan to Middlesbrough was extended for an additional season. He made his season debut on 24 August against West Bromwich Albion, and provided an assist for the winning goal of the game. On 2 October, he scored his first goal of the season against Ipswich Town.

On 8 August 2019, Bešić was sent on a season-long to Sheffield United. He played his first official game for the team in EFL Cup against Blackburn Rovers on 27 August. Two months later, he made his league debut away at West Ham United. Bešić scored his first goal of the season in an FA Cup tie against Millwall.