Born in Stenhousemuir, Scotland, Leishman played for St Mirren before Liverpool signed him for £10,000 on 20 November 1959.

His first goal came in the 11th minute of a league match at The Dell on 24 August 1960, unfortunately it turned out to be a consolation as the Saints hit 4 past the Reds backline.

Tommy had a great touch but he was also a driving force in the middle of the park, he settled in well and appeared 17 times in the remainder of the 1959–60 season.

The following year they finally achieved their goal as they finished eight points clear of second placed Leyton Orient with Leishman missing just one game.

He failed to secure a regular place in the Hibs first team and he joined Irish League side Linfield as their player-manager.