The League One club appointed Kelly last week but the 35-year-old from Northern Ireland has since had to refute claims that he is a supporter of the IRA.
A club statement on Monday afternoon read: “Oldham Athletic AFC are aware of threats made against individuals connected with the club.
“These matters have been passed to the police and the club will be making no further comments on this subject.”
Some fans have threatened to boycott Oldham’s matches following the decision to employ Kelly.
The Latics caused controversy in December when they were in talks to sign convicted rapist Ched Evans, before pulling out of the deal after a number of sponsors threatened to withdraw their backing.