Spence used his Twitter account tonight to say: "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."
Police Scotland confirmed Spence had called for advice and an officer had spoken to him, but said the force had not been formally involved in any investigation.
Earlier this week, The Herald revealed that the BBC had apologised after 400 Rangers fans complained over a comment Spence made.
In a discussion over the appointment of non-executive directors to the Ibrox board, Spence said on BBC Radio Sportsound: "John McClelland who was the chairman of the old club, some people will tell you the club, well, the club that died, possibly coming back in terms of the new chairman."
BBC Scotland denied Spence had claimed Rangers had 'died' but apologised to "those who were offended by this".
Paul Holleran, NUJ organiser for Scotland, said: "The BBC has offered its total support to Jim. We don't condone the threats and disgusting emails and texts that he had received and find it totally unacceptable that a journalist has been treated this way."
Spence later tweeted: "Thank you for your overwhelming support, a huge thank you to the NUJ and to many in the media and blog and zine movement."