The incident comes after the sports presenter found himself engulfed in a row with Rangers fans.
Mr Spence used his Twitter account to thank the police for their help following the incident, which occurred yesterday afternoon.
He wrote: "Thanks to @TaysidePolice for a reassuring call there after an incident earlier today. This is getting out of hand."
He added: "Was out for a walk with my wife...Abuse on twitter is one thing but when people start on you in the street with your wife that's another entirely."
A spokesman for Police Scotland said Mr Spence phoned his local police station for advice and an officer later returned his call, but police were not formally involved in any investigation into the incident.
This week, it was revealed the BBC apologised after 400 Rangers fans complained when Spence spoke of the Rangers "old club" on BBC Radio Sportsound, saying: "Some people will tell you the club, well, the club that died."
BBC Scotland denied Spence claimed Rangers had "died" but apologised to "those who were offended by this".
Meanwhile, the Ibrox side confirmed that it has signed a partnership deal with E-lites electronic cigarettes. E-lites are already in partnership with Celtic to sell green-tipped electronic cigarettes at Parkhead.