He was originally hit with a $10,000 fine, 60 per cent suspended, as well as a suspended one-match ban after initial investigations.
It has since been alleged that the Panthers halfback was not at home for the entire day in question, prompting NSW deputy police commissioner Gary Worboys to look into the matter again.
“There will be some further inquiries made into that over the coming days and if it’s appropriate to take action against that person, that will occur,” Worboys said.
“It’ll be a local Police Area Command investigation.
“They handled it previously and we’ll look forward to the outcome of that in the coming days.”
Panthers teammate and Cleary’s housemate, Tyrone May, was fined $15,000 by the NRL for his role during the league’s investigation.
Penrith had hoped to put the saga behind them after copping the NRL punishments for both players on Wednesday.
Coach Ivan Cleary, the father of Nathan, said it was out of character for his son, who has had an unblemished record off the field since his first-grade debut in 2016.
“He is very remorseful and he’s paying the price,” Cleary told the club’s website.
“I actually think it could be the making of him, I really do. He’s still a young man.
“He knows he’s done the wrong thing and he’s hurt people.
“In my experience, sometimes the best leadership lessons are done through practising the hard way.
“There’s not many great leaders who don’t make mistakes, who can be vulnerable, do the wrong thing and make bad choices.”