The 31-year-old made his debut in the fixture and was criticised by Celtic manager Neil Lennon for awarding Rangers a penalty in the second half of the 3-1 win for the visitors.
The official reportedly answered a number of phone-calls at his Lanarkshire home with the caller threatening to "go after him and his wife and children".
It is understood that Collum reported the calls to the police but turned down the offer of protection outside his home.
SFA spokesman Darryl Broadfoot told the Daily Record: "Willie received a number of threatening calls at his home on Sunday night, one of which was taken by his wife.
"This kind of behaviour is abhorrent and has no place in football."
The incident can only de-motivate further the several top officials who are believed to be considering their futures due to the intense scrutiny they are being placed under.
Assistant referee Steven Craven resigned on Monday after being caught up in a furore over a decision to deny Celtic a penalty at Dundee United last week.
Meanwhile, Celtic assistant manager Johan Mjallby hit out at his players' performance against Rangers, telling the Scottish Sun: "I was disappointed with how we lost the initiative in the second half.
"It was almost like boys against men in the second half.
"Rangers were very effective, but we did not get a shot on target in the second half."
Mjallby, however, refused to single out any of his players, saying: "We do not comment on individuals.
"We win and lose as a team, but we gave away goals - and the match - too cheaply."