The pair, both 21, are clients of Jorge Mendes' GestiFute agency, whilst also represents the likes of Cristiano Ronaldo and Jose Mourinho.
Mendes' links with Championship club Wolves came under scrutiny last season, with Portuguese publication 'A Bola' now reporting that Forest hope to follow suit in luring some of Europe's top young talent to the English second division.
Midfielder Carvalho made seven league appearances for the Portuguese side last season and started their final Champions League Group Stage game against Basel, whilst Gonçalves, who can play anywhere across the front line, laid on two assists in seven Liga NOS games as well as making four startsin the Champions League.
Should the deal for Carvalho go through, it would see the 'Trick Trees' smash their transfer record by some £7.5m, eclipsing the £5.5m they paid Peterborough for striker Britt Assombalonga back in 2014.
Both players are Portugal U21 internationals and their arrivals be a major boost as Aitor Karanka looks to build a side capable of promotion.