The City ground boss worked with the 31-year-old during their time together at Deepdale and has already failed with two offers as he attempts a reunion.
Rumours at the end of June suggested that Forest had made a third approach to their Championship rivals and Davies has confirmed the speculation.
Davies told the Lancashire Evening Post when asked about McKenna: "We have put in another improved offer to Preston, believed to be in the region of £750,000.
"Paul McKenna is a player I admire greatly and I believe he is the right age for this club because of his experience and what he has done.
"He is a very fit man, a great professional and a great talker in the dressing room and on the pitch.
"He is an outstanding player and we will do all we can to bring him to Nottingham Forest because he is a player we need.
"I think he probably needs a fresh move for his own career."