Aleksic was signed by First Division side Tottenham Hotspur in December 1978 for a fee of £100,000. The club posted a 14th-place finish in the 1979–80 season under the stewardship of Keith Burkinshaw. He faced competition between the sticks from Barry Daines and Mark Kendall. In one tie against Manchester United, notorious 'hard man' Joe Jordan broke his jaw in what was officially described as an 'accidental collision'. The injury left midfielder Glenn Hoddle to take Aleksic's place in goal for the rest of the match; despite this setback, Hoddle was untroubled in goal, and Spurs won 1–0 with Osvaldo Ardiles scoring the winning goal. The White Hart Lane club made a top ten finish in 1980–81, and also lifted the FA Cup with a 3–2 victory over Manchester City at Wembley. Aleksic featured in both the original tie and the replay, which was won with a brace from Ricardo Villa and a scruffy goal from Garth Crooks. However, he soon lost his first team place to Ray Clemence, and was on the bench for the 1981 FA Charity Shield draw with Aston Villa. Spurs went on to finish fourth in 1981–82, but despite being an FA Cup winning goalkeeper just one year previously, he dropped back into non-League football with Barry Fry's Alliance Premier League side Barnet in 1982, following a brief loan spell at former club Luton Town. He later emigrated to South Africa, and played for Wits University, before taking up employment at the Golfer's Club in Johannesburg.