Mr Spence provoked the ire of Rangers fans after making the comments on the Sportsound radio show about attempts to end a boardroom civil war by appointing four additional non-executive directors, including former Ibrox chairman John McClelland, bus tycoon Sandy Easdale, former director Paul Murray and accountant Frank Blin.
It comes three months after the BBC Trust ruled that BBC Scotland breached its guidelines on accuracy in reports about the financial collapse of the holding company running Rangers.
The Trust upheld two complaints that the BBC had been wrong to use the terms "new" and "old" club.
The BBC says Mr Spence said during the broadcast: "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." A clip of Mr Spence posted on YouTube further shows he described Rangers as "the old club, the old regime".
The Rangers Standard forum said: "It appears that (Jim Spence) has disgraced himself again tonight. Dishonest baiting of RFC fans his only contribution."
BBC Scotland, which carried out a review of Sportsound, has apologised to fans. A spokesman said: "Jim Spence did not state that the club had died but, in the context of a discussion about former board members joining the club, commented that there are some people who hold this particular view. I do agree that the attributed comment could have been better phrased within the live piece and I am sorry if you were offended by this."
Mr Spence, a Dundee United fan, hosts Sportsound twice a week.