The former Crystal Palace and Sheffield Wednesday striker, who works as a summariser for BBC television and Radio 5 Live, told jokes about Terry and Bridge and repeated an abusive terrace chant about Terry's conduct, which suggests it would be unwise to leave your wife close to the England and Chelsea captain.
Bright has 5,984 followers of his Twitter account, which mostly includes reports of his BBC assignments.
What a twit: Mark Bright's Twitter page containing his apology
He has recently been working at the Africa Cup of Nations in Angola.
The posting of the chant included poking fun at Terry's penalty-taking skills at the 2008 European Cup final, where he missed in the shoot-out against Manchester United. Terry has said it is the worst moment of his playing career.
Bright, 47, later deleted the chant and wrote: 'After 4,362 tweets, 5984 followers, apologies to JT & Bridgy if offended, error of judgement from me, thank u all 4 the banter, im gone.'
The BBC will take no action but a spokesman warned: 'Personal accounts that have been set up by an individual on a social networking site are the responsibility of that individual.'
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