Terry, who left Villa in July, is reportedly ‘interested’ in managing Forest.
The Sun claim Terry and former Sheffield United boss Chris Wilder are both watching the situation at the City Ground with Chris Hughton under pressure.
Forest are bottom of the Championship ahead of Sunday’s game with Cardiff City having picked up one point from four matches.
Terry spent time on loan at the Reds at the turn of the millennium and could begin his managerial career in Nottingham.
The former Villa defender arrived in B6 in 2017 as a player before becoming one of Smith’s assistants three years ago.
I’m always waiting, but the idea was to step away and spend the year with family and see what comes up,” Terry said in an interview with Sky Sports earlier this week.
“You know what it’s like, we’re a month into the season. I’m sure jobs will come up and I’m there and waiting and doing what I need to do in the background, but if not, I’m enjoying my time with the family.
“I love Villa as a football club but I think everyone knew when I went in there I wanted to be a number one and that’s very much at the forefront as I speak.”
Forest were expected to challenge for promotion this season having thrown away a chance of the play-offs in 2019-20 and a disappointing season in the most recent campaign that saw Sabri Lamouchi sacked and replaced by Chris Hughton.
They are propping up the second tier with their sole point coming in a draw with Derby County last time out.