The 22-year-old is said to have committed the first offence at 11 p.m. GMT on Saturday, Nov. 16, before doing so again at 2 a.m. the following morning.
As it was an international week, the Bhoys had no fixture that weekend, and Forrest was not called up to face the United States as part of Gordon Strachan's Scotland squad.
The report goes on to say that Forrest is believed to have been in Elliots Bar in Prestwick at the time the offences occurred.
Despite the revelation, a Celtic spokesperson confirmed Forrest has been released and remains a part of Neil Lennon's Celtic squad to face Milan on Tuesday night.
According to the Daily Record, Police Scotland said:
A 22-year-old man is subject to a report to the procurator fiscal in connection with two alleged public indecencies in Prestwick on Saturday November 16 into Sunday November 17.
Forrest has struggled with injury this season, making just six league appearances and scoring two goals.
However, the wide man has started in both of his side's last two European fixtures, scoring in the 2-1 win over Ajax a month ago.
That victory remains the only thing keeping Celtic afloat at this stage of their European campaign, and a loss to Milan would eradicate any chances the Scots have of advancing to the round of 16.
In that scenario, a future in the Europa League would rest on Celtic being able to grab points from their trip to Barcelona in December, assuming Ajax don't claim any more points of their own.
For now, Lennon has the youngster as a readily available asset to use against the Rossoneri, in what is a vital game for Celtic's Champions League hopes.