The club reported the death threat against manager Darren Kelly, who is from Derry, to police after receiving a letter on Monday.
Police said a 31-year-old man was arrested on suspicion of making threats to kill.
Two men, aged 27 and 42, arrested on suspicion of making threats to kill on Tuesday have been bailed until 15 June.
35-year-old Kelly, who was appointed as Oldham manager on 5 May, has had to deny claims he supported the IRA after he reportedly called Sinn Fein politician and Deputy First Minister Martin McGuinness a "legend" on Twitter.
He made the comments after Mr McGuinness retweeted a photo of him meeting the Oldham boss's mother at an event marking the 43rd anniversary of Bloody Sunday.
Kelly, was a defender at Derry City, Carlisle, Portadown and York then coach at Sunderland.