McGlennan will replace Vincent Lunny after a handover period next month.
He will assume responsibility for determining which players, coaches and other officials should face SFA disciplinary charges.
McGlennan joins from Possilpark-based law firm Penmans, with whom he has been a partner since 1998, two years after he joined.
An SFA statement read: "He is a solicitor advocate who has conducted criminal litigation at all levels of the court structure, including the Appeal Court and the Supreme Court.
"Tony has a keen interest in football, having played at amateur level for a number of years. He is currently a volunteer coach to young footballers and holds coaching certificates in the Scottish FA's youth coaching pathway."
Andrew McKinlay, the SFA's director of football governance and regulation, added: "As everyone knows, this is a hugely important and challenging position in Scottish football, one which comes with a great deal of pressure, and I am absolutely certain that Tony will be an asset to the organisation and the game.
"I would also like to thank Vincent for his excellent work during his three years with the Scottish FA."