Sport says Real president Florentino Perez has offered to include Vinicius in a bid which would be worth €250m.
Florentino has decided to make the move with coach Zinedine Zidane having no great confidence in Vinicius.
Real had tried to include Gareth Bale, Luka Modric and Isco in previous offers, but PSG weren't interested. However, there is a greater confidence a proposal including Vinicius will get a deal over the line.
With Eden Hazard's arrival, Florentino accepts they can afford to lose Vinicius should Neymar arrive in his place.