The 36-year-old striker, who was recently released by Bolton Wanderers after 10 years with the club, held talks with Preston earlier this week.
Grayson hopes that he has done enough to persuade Davies to join the League One club.
"I'd like to think we've got certain things that will be in our favour," he told BBC Radio Lancashire.
"There might be other things that are not, but hopefully we've ticked enough boxes to suggest this is the right football club for him at this stage.
"If Kevin signs for us, it will be for personal and footballing reasons rather than financial reasons."
Championship sides Sheffield Wednesday and Wigan Athletic have also been linked with a move for Davies.