Officers confirmed they have received a complaint against Father Gerry Nugent, who died in 2010.
It comes as BBC Scotland reported allegations on Friday from a woman who claimed she was abused by him when she was a teenager.
A Police Scotland spokeswoman said: "We can confirm we have received a complaint and enquiries are ongoing."
The matter is a live investigation, she confirmed.
Fr Nugent was a key witness in the trial of serial killer Peter Tobin, who was convicted and jailed for life in 2007 for raping and murdering student Angelika Kluk.
Miss Kluk's body was found in September 2006 under the floorboards of St Patrick's Catholic church in Glasgow.
During the trial, Fr Nugent - who was the priest at the church at the time of the killing - claimed to have had a sexual relationship with Miss Kluk.
The shamed priest was asked to resign from his post at St Patrick's.
The Catholic Church said it is aware of complaints against Fr Nugent, some going back 40 years.
A spokesman said: "The crimes described in these allegations are appalling.
"The Church, its bishops, priests and people are truly sorry for what has happened.
"The Archbishop has met the survivor interviewed by the BBC and apologised in person and on behalf of the Church for what she has gone through."
He added that the Church was "ready and willing" to co-operate with the police and other authorities to assist survivors.
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