The Italian spent one season at Ibrox in 1997-98, before a successful 13-year spell in his homeland with Milan.
Since retiring, Gattuso has had jobs in charge of Sion, Palermo and Crete, and was thought to have applied for the job at Hamilton Acamedical, following Alex Neil's departure to Norwich City.
Speaking at a tribute match for former Rangers defender Fernando Ricksen on Sunday, Gattuso stated his wish to manage the club, who are currently in Scottish football's second tier as they work their way back towards the top flight following financial ruin.
"My dream is to maybe one day to have the chance to work for this club," he said.
"I played today, but manager is a different job. I am sure 100 per cent that this is a big club."
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