As it stands, the Italian is due to return to his parent club Manchester United at the beginning of February.
However, the Peterborough boss is hoping that his father Sir Alex Ferguson will allow the 21-year-old to extend his stay at London Road.
"He is, by definition of where he came from, a very good footballer and he loves being at this club," said Ferguson Junior. "We only have him for a month at first, but I will definitely try and extend the loan."
Petrucci joined United from Roma in the summer of 2008.