Career
Early career
Born in Titograd (currently known as Podgorica, the capital of Montenegro) he lived as a youngster in Serbia, in the city Lučani, where he played for the youth team of the local club FK Mladost Lučani. He later joined FK Budućnost Podgorica where he made his senior debut playing in the Montenegrin First League in the 2008-2009 season. He also had a half season loan spell with FK Mladost Podgorica.
Red Star Belgrade
Vešović was scouted and purchased by Red Star Belgrade from FK Budućnost Podgorica in June 2010.
Torino
On 30 January 2014, he signed with Italian side Torino on a free transfer, after a couple of days on trial under the manager Giampiero Ventura. He made his official debut for the Granata on 9 March 2014 during the trip to Milan against Inter, in which he was a starter, playing the full 90 minutes, lost 1–0.
By the end of the 2013-2014 season he made four appearances for Torino. = Rijeka (loan) In summer 2014, he moved to Rijeka in Croatia"s Prva HNL on a one-year loan.
Spezia and Rijeka
On 8 August 2015 he was sold to Spezia in Serie B for an undisclosed fee, but immediately returned to Rijeka on another season-long loan.
After having been part of the Montenegrin U-19 team, Marko Vešović has been a regular presence in the Montenegrin U-21 team since 2009. On May 20, 2013, Marko was called up for Montenegro national football team for the first time.