Davies made a return to the East Midlands club on Thursday, almost 20 months after he was sacked by the old regime.
Davies has replaced fellow Scot Alex McLeish as manager two days after the former Birmingham and Aston Villa boss' brief spell in charge ended.
After steering the Reds away from relegation following his appointment in January 2009 and then masterminding successive Championship play-off campaigns during his first spell, Davies insists he has "unfinished business" at Forest.
The 48-year-old - who has been given a three-and-a-half year contract by the club's Kuwaiti owners, the Al-Hasawis - officially started work on Monday and was clear in his objective as he addressed the media for the first time.
"I've said before that I feel the last time I left was a little bit too early - there was unfinished business," Davies said.
"I genuinely believe that, and I've always said that I would love to come back to this football club and finish what we started.
"I think that's got to be Premier League football. I think we've got to try to take this club back to where everyone thinks it belongs, I have no doubt about that.
"That is certainly the ambition of the owners and the majority of the people at this football club, and it is certainly the ambition of myself and my staff.
"I am very much looking forward to that challenge. I have always said that I would like to come back and finish the job off one day, although I never thought that my very next chapter would be back here at Nottingham Forest."