East Stirlingshire FC's chairman expressed "deep regret" for the distress caused by comments on the social network site of Brett Molloy.
The under-19 player says he was the victim of a hacker and has filed a complaint to police.
In a statement issued through lawyers, Molloy said: "I have no comment to make at this time whilst club investigations and police investigations into my complaint are ongoing."
Strathclyde Police said it was investigating the remarks. A force spokeswoman said: "We can confirm a complaint has been received and inquiries are ongoing."
The force was unable to confirm whether or not a complaint had been received from Molloy in relation to his Twitter account allegedly being hacked.
The Ibrox Disaster claimed 66 lives on January 2, 1971 when fans were crushed on stairway 13 of the stadium at the end of an Old Firm game.
East Stirlingshire chairman Tony Ford published the apology on his club's website.
It read: "I have been contacted by a number of people regarding posts made on social networking sites by one of our under-19 youth team players and I would like to thank those people for bringing this matter to my attention.
"Having read these comments I share your sense of shock and revulsion.
"Furthermore, I am horrified that a player connected with our club can express such opinions.
"I am sure all Shire supporters will join me in condemning these words.
"On behalf of everyone at East Stirlingshire Football Club I would like to apologise for the actions of this individual."
The statement went on: "The player concerned has already been spoken to by club staff and has been told he will be the subject of a disciplinary procedure which will begin today."
All of the club's players were spoken to about the use of social networking sites at the start of the season, Mr Ford said.
The statement was replicated on the Rangers FC website.
A foreword from the Glasgow club stated: "It is truly abhorrent that anyone could see fit to mock such a tragedy.
"Rangers condemns the sickening comments made and we note East Stirlingshire have issued a swift apology to everyone connected to the club."