Carvalho featured sporadically for the Encarnados last season, but he is a Portuguese Under-21 international and a potentially ambitious investment from Evangelos Marinakis, who is backing former Middlesbrough boss and current Forest incumbent Aitor Karanka to the hilt.
The Telegraph also reports that the midfielder's Benfica teammate Diogo Goncalves is flying into England on Wednesday night to complete a season-long loan to Forest.
The previous regime of Fawaz Al Hasawi left Forest in a bit of a financial mess and they're having to cut their cloth accordingly, so these moves are even more of a coup because of the low wages being paid to both new acquisitions.
The fee for Carvalho may rise to around £15m, almost three times the £5.5m that Forest paid Peterborough for Britt Assombalonga in 2014 - their previous record signing.
Forest finished 17th last season but there were encouraging signs under Karanka and the Spaniard hopes that up to eight new signings this summer can turn the club into promotion challengers.
The next man to arrive at the City Ground could be defender Alan Hutton, who is close to agreeing a two-year contract with Forest as he departs the sinking ship at Aston Villa.
Watford goalkeeper Costel Pantilimon, who spent the second half of last season on loan at Forest, is also on Karanka's shopping list, as well as another forward.
Room on the wage bill will have to be cleared to make way for more arrivals, with Eric Lichaj, Jamie Ward, Michael Mancienne and Barrie McKay all expected to leave.