Kelly was appointed boss at the League One club last week but the 35-year-old from Northern Ireland has since had to refute claims that he is a supporter of the IRA.
Police were called to Oldham’s Boundary Park stadium on Monday morning after the club received the threatening letter.
Two men, aged 27 and 42, have been arrested on suspicion of making threats to kill and remain in custody, Greater Manchester Police said.
Detective Inspector Jason Byrne, said: “The letter sent to the club made some extremely serious threats to a number of individuals and we are treating this matter with the utmost seriousness.
“We are also aware of equally vitriolic comments that have been made on social media and this too forms part of our investigation.
“We have made two arrests and anticipate more. Another man knows we are keen to speak to him and I would stress that it is in his best interests to come forward.
“I would also ask anyone who has information about the letter or those behind it to contact us.”
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.