The 20-year-old even contacted both clubs personally to make them aware of his decision, according to the Daily Record.
Campbell, who has impressed in recent months for Stoke City, is out of contract at the end of the season, leading a number of clubs to show their interest.
The Old Firm would have been able to take advantage of the cross-border compensation rules for young players coming from England. Celtic brought Moussa Dembele to Scotland using it, as did Rangers with Joe Aribo.
The Scottish champions were reported to be the first to show interest via transfer chief Nicky Hammond before Rangers joined the race.
The Ibrox side were understood to put across a very attractive offer for the striker.
Despite long considerations, Campbell has opted to stay in England.
Premier League side Sheffield United are very keen on the player, while Stoke City have not given up hope of convincing the player to stay.