The owner of Forest, Evangelos Marinakis, is said to view Silva as an ideal choice to manage the team at the City Ground.
However, any potential move for Silva is unlikely to happen until the summer, as it would likely involve a significant compensation fee.
Silva had previously resisted an approach from Benfica to stay at Fulham, where he has been managing this season.
Meanwhile, Forest's current manager, Ange Postecoglou, will be speaking with Marinakis this week to discuss his situation, as he has yet to win any of his first seven games in charge.