According to reports in the French media, Forest are keen to land the 32-year-old Senegalese midfielder to their growing ranks even further, and could beat several Turkish clubs to his signature.
Kouyate left Palace after failing to agree a new contract with the Selhurst Park club, as he looks to collect one big final pay day with his career drawing to an end.
Nottingham Forest have been on the hunt for some added steel in their midfield, having missed out on Paris Saint-Germain midfielder Idrissa Gueye who is on the verge of a return to Everton. The experienced Kouyate certainly fits the bill.
Kouyate scored 15 goals in 147 appearances for West Ham, and wore the Palace shirt 141 times. He played 27 times in the Premier League last season, so was clearly an important part of Patrick Vieira's team.
Forest manager Steve Cooper admitted before the opening day away defeat to Newcastle on Saturday that the club were still searching for new faces to add to the 12 he had already brought in.
Jesse Lingard and Dean Henderson headlined the those arrivals, with the club's transfer record also being broken on Taiwo Awoniyi, but Kouyate has already spent a long time in the Premier League and will be able to pass on his vast experience.
Cooper said: "I think there will be more ins and outs. A number I don't know.
"It's not a straightforward process in bringing a player in, I don't know in terms of numbers.
"My focus is on the players that are already here, I imagine we're getting linked with all sorts of names, maybe some truth in them and some there will be no truth at all.
"My focus is on the guys who are coming into training every day and preparing for the next game, because they deserve 100 per cent attention."