Police have confirmed they have received a complaint after referee John Beaton's contact details were leaked online.
Press Association Sport revealed on Friday night the 36-year-old has been targeted by trolls in the wake of last weekend's Old Firm clash.
A spokesperson for Police Scotland said: "We can confirm that a complaint has been made to police regarding texts and calls received by a 36-year-old man. Police enquiries are ongoing into this matter."
Parkhead chiefs have called on the Fifa-listed whistler to explain why he did not take action against the Colombian following three flashpoints with Hoops players.
But he was forced to call in police when his work mobile phone number was leaked on social media.
The official has since been bombarded with dozens of threatening and abusive messages in WhatsApp.
Scottish FA chief executive Ian Maxwell condemned the threats made to Beaton.
In a statement on the SFA's website, Maxwell said: "It is with deep dismay that in the last 24 hours, we learned of one of our match officials having had threats made against him and his family.
"We are in close contact with those involved and with Police Scotland to ensure that they are safe and that those responsible are identified and held accountable for their actions.
"This is not the first time in recent weeks that our match officials have been targeted. Another referee was allegedly threatened and assaulted at a lower-league game prior to Christmas. These incidents are isolated but are they are unacceptable and extremely concerning when they occur.
"Our match officials take huge pride in their work but it is inevitable and accepted that some of their decisions will be viewed differently and debated by others.
"When differences of opinion, however, manifest as threats to our officials or their families, a line has been crossed."
Beaton took charge of Ayr United's clash with Falkirk at Somerset Park on Saturday afternoon.
Celtic, meanwhile, have requested a meeting with Maxwell after compliance officer Clare Whyte decided against taking retrospective action against Morelos.
The Light Blues' top scorer appeared to kick out at Hoops skipper Scott Brown, stand on Anthony Ralston and flick out his hand at Ryan Christie's groin area during the victory on December 29 which took Steven Gerrard's team level on points with Brendan Rodgers' Ladbrokes Premiership leaders.
However, Whyte was unable to take any further action after Beaton confirmed he saw all three incidents in their entirety during the game.
Celtic released a statement which expressed "surprise" at the decision and will look to meet with Maxwell and the SFA's head of refereeing, John Fleming.